Aegolius13

That's been my experience -- bigger hinges can move bigger loads, and seem to do it more smoothly.

Yep, this is called a "cosine loss."  The amount of wasted thrust is proportional to the cosine of the inclination (0% wasted if the engine is pointing exactly where you want to go, 100% wasted at 90% inclination).  The effect on delta-v would be the same, since cosine losses decrease the amount of usable thrust generated per unit of fuel consumed.  It's akin to lowering the ISP of your thrusters.
The above should hold true for one-direction thrusters or regular engines.  I've never really delved into it with the multi-direction thruster blocks, but I think it would work the same way, since using the additional thrust directions (which are pointed even more in the wrong direction) would not be expected to improve efficiency. 
I'm not a MechJeb user, but I believe it provides some data on cosine losses. Don't know if this includes info from RCS, though.
(My very first post on this forum was on the same topic, so you're in good company!)

If you're looking for specific advice on your ship, it would be very helpful if you could post a screenshot.  There are a ton of potential issues that can limit your range, but so it's hard to which advice might or might not pertain to your project.  
That said, funnily enough, I was just starting to work on an "intermediate player tips" post with just such general advice.  It's still a work in progress but I'll post some general content below.  If you're still figuring out delta-v and the Rocket Equation, for the discussion below just keep in mind that more delta-v means longer effective range. 
Unfortunately, there is no simple answer to this question, because it depends on a lot of other variables.  If you make a very small, efficient Mun lander and fly it perfectly, you can get away with a small rocket to start with.  But if you bring a lot of stuff (either because the mission requires it or do to design inefficiencies), your rocket will need to be bigger.  
I would very much recommend that you get used to designing from the top down.  Start with your final stage and figure out what you need on it to accomplish its goals.  Then figure out what the next stage needs to get your final stage where it needs to be, and so on and so on, until you've gotten all the way down to your launch configuration.  
I'd also suggest you keep working on delta-v, and playing around the delta-v readout (or a similar mod tool, like Kerbal Engineer Redux) until you're comfortable with the basics of what it's telling you.  For instance, you may have seen that it typically takes 3,000 to 3,500 m/s to make Kerbin orbit.  It's very helpful to be able to take that fact, and use it to see what kind of a launch configuration will give you that much delta-v. 
As far as your question about thrust, are you also familiar with thrust-to-weight ratio?  That's a more important number than total thrust.  TWR is another data point you can see in KER, or fudge by looking at the g-meter.  But as far as how much thrust you need, you probably want your ship to start with a TWR somewhere between 1.3 and 1.8.  Later stages can have progressively less TWR, but It'd keep it above 1 to start with until you're in orbit.  From that point on, anything over 0.5 should suffice.
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The Rocket Equation: Live it, Learn it, Love it.  Do you want your rocket to go farther?  It seems like a complicated proposition, but thanks to the Tsiolkovsky Rocket Equation, it’s actually a pretty simple matter.  The equation solves for the delta-v a rocket can achieve using three, and only three variables:
1.       Dry Mass: the mass of everything in your rocket other than fuel
2.       Wet Mass: the mass of the fuel in your rocket.
3.       Specific Impulse: the efficiency of your engines.
The first two can even be condensed into a single number called the mass fraction (the ratio of dry mass to wet mass), giving you only two variables to work with.  So if you want a rocket to go further, the equation gives you a few general guidelines:
1.       Decrease Dry Mass: this is arguably the most important point, and one I’ll touch on in greater detail below.  But cutting unnecessary fat can pay enormous dividends in terms of delta-v.
2.       Increase Wet Mass: that’s a fancy way of saying “add more fuel.”   Simple enough, as long as the rest of your rocket can handle it.  But be aware that adding fuel runs up against diminishing returns pretty quickly, because that extra fuel equals extra dead weight up until the point it’s burned.  This is the core of the “Tyranny of the Rocket Equation.”
3.       Increase Specific Impulse: this just means selecting an engine with favorable ISP numbers for the task at hand.  Again, simple enough on the surface.  But (again as discussed below) ISP is just one of many factors you’ll have to balance, and can be a bit of a trap if over-prioritized.
There’s one more range factor that’s not expressed in the classic Rocket Equation, but obviously has a large effect on your ship’s range.  That’s staging, and while it’s an important topic, I’m not going to get too far into the theory here.
 
Keep it Simple: This goes back to the “reduce dry mass” point above.  It is almost impossible to overstate how large an effect dry mass can have on your ship’s delta-v.  I don’t want to get too technical, but a very significant fraction of your ship’s delta-v comes when your fuel level is low, at which point the rocket is very light, and every unit of fuel burned can produce a lot of acceleration.  Keeping dry mass light really helps amplify this effect.  This is actually the major reason why real-life rockets can achieve much greater delta-v per stage than the KSP equivalents – their fuel tanks and engines are much lighter, giving you much better mass fractions.
Reducing dry mass can pay other dividends as well.  You will get better TWR out of your engines, which at a minimum can improve maneuver efficiency, but can sometimes allow you to switch to lighter (or fewer) engines – further reducing dry mass in a virtuous cycle.
So with that as background, it’s always good practice to approach a rocket with the question of “what can I cut?” rather than “what can I add?”  
 
 

You need all the normal utility stuff to keep the station running -- probe core, electric generation,  batteries, antenna, attitude control (RCS and/or reaction wheels), etc.
You probably want some crew modules, just in case. 
As mentioned above, it's common to store fuel. 
I'd add every science instrument you have.  Even if you've already exhausted those experiments, you may periodically get a contract to perform science at the station's location.  If you do, free money!  
On that note, it's good to add a science container.  If nothing else it simplifies the act of collecting science from docked ships.
The Mobile Processing Lab is a bit controversial.  It can get you a lot of science, but it's fussy and confusing to use, and overpowered if you do get it working right.  But in addition to the main function of generating science, it can level up your crew without retuning to Kerbin.  I usually include it for this reason, and just for roleplaying I suppose. 
You'll definitely want docking ports.  I'd add quite a few, add different sizes just in case, and try to arrange them so that docked ships pr expansion modules won't run into each other. 
I'd add some propulsion to the station. Occasionally you may need to do a bit of active rescuing, or get a contract to move the station to another orbit.  I tend to radially mount engines, to keep the ends clear for docking ports.  You probably don't need much thrust. 
 

I think your first decision is whether you want some throwaway thing to just complete the contract, or add something that's long-term useful to your fleet.  As you say, even in the worst case, you can send the thing back to orbit and add the LF to a fueling depot.  
If you want something useful, maybe a nuclear-powered  mothership that you can use for the interplanetary missions you're planning.  I.e., a big ship with lots of crew accomodation, a science lab, and capable of docking and refuel a lander.    Technically, you don't even need to have landing legs, as long as you can get it to land stably, and that's not too hard to do on Minmus with anything.  But you could always launch it with some detachable girder things with wide landing legs, and then decouple them after you complete the contract.  
As far as something that will STAY on Minmus - that's tougher.   I guess you could build a fuel depot-type thing, and supplement it later with a mining rig.  But at least with my playstyle, I'd be pretty unlikely to actually use such a thing.   

Here's a single-stage (after reaching LKO) Mk 3 lander I've been using lately.  It can make a full circuit to the Mun, Minmus, interplanetary space and back on one tank.  This version requires a mod to switch the fuel tank/adapter to LF only, but you could do a similar (uglier) design with a short Mk 3 LF tank.  The command pod is also modded, but you can swap in pretty much any pod/cabin on top, or just use a probe core and some reaction wheels.  If you launch it in a fairing, no need to worry about aerodynamic design.
This can either be refueled in space and reused indefinitely, or modified with chutes to land back at Kerbin.
 

 
If you'd rather not go nuclear, you could probably do an LFO version of this with Terriers on the nacelles.  And then maybe radially attach a couple detachable 2.5m nacelles with Poodles for extra range. 

Nice design!  No mean feat to get an orange tank to orbit.  A couple thoughts:
-That 2.5 nosecone is fairly draggy (or at least used to be; not 100% sure still the case) .  You could go with some adapters to taper your front down, and put a 1.25m advanced nose cone,  small nose cone, shock cone, or even a fairing you never pop in front.  
-Do you really need those radial air intakes?  If you're borderline, you might be able to eliminate if you add an extra shock cone per previous point.  
-You might consider the Big S delta wing.  It has the same lift-weight ratio as the modular wings, but comes with free LF storage.  It also has good temp tolerance, and tends to be less prone to breakage than the modular wings.  If you need a little more lift, you could look at the wings strakes too, use a bigger canard, or just slap a few modular wings on the back of the Big S.
-Are those multiple linear RCS ports?  You could try switching to Vernors.  That might save drag or at least simplify your fuel situation.  Or ideally, try to make it with no RCS at all.
-You might have a little more roll and yaw control authority than needed, though it is a big plane so maybe not.  You can also get the most of your pitch authority by putting it as far from the COM as possible.  That probably means putting big canards as far forward as they'll go.
-I have never really used it, but vaguely remember hearing some hate on the Space Shuttle-style engine mount for having high drag and mass.  If it does seem draggy, you could taper down to 2.5m and use one of the multi-couplers to mount your engines. 
-Those Mk.2 fuselages you have on the side also have a bad rap for being draggy.  You could look at doing two 1.25m stacks on each side in order to keep the same number of engines.  

If you're doing a single pod mission (i.e., NOT Apollo style), I think the command pod is better overall.   It's got better aerodynamics for launch, more SAS torque, more heat resistance,  and is cheaper to boot.  While the lander pod is lighter,  the difference is not that big. 
If you're doing an Apollo style mission,  the best option is probably to use each part like the real thing.  Land on Mun with the lander,  renter Kerbin with the command pod. More generally,  i like the lander for missions that stay in vacuum,  and the command pod for stuff that includes reentry.

I played around a little with the plane.  Nice design!  Think it's bigger than any plane I've gotten to orbit.  I don't see a lot of changes to up your range, but here are a few.  The general idea was to try to lose a little dry weight, lose a couple Rapiers, and add at least one nuke. 
-You might be able to lose those precoolers (or replace with liquid fuel fuselages), especially if you can cut down on the number of Rapiers by a little, and/or put a shock cone on the front of the plane.    Despite the part description, the precooler is just an intake and does not do anything useful if you have enough air from elsewhere. Shock cones can feed a fairly ridiculous number of engines.  They might sputter a bit until you hit 10 m/s or so on the runway, but you should be good from there.
-Those Big S elevons near your middle engine clusters are not doing a lot to help you pitch.  They work like levers, so when they're near the center of mass, you get very little leverage.  One thing you could try is to leave just one set of elevons there (as far outside as possible), set them for roll control only, and then beef up the number or size of control surfaces you have at the forward and back ends of the plane.  This might allow you get the same control authority but for lower part count, mass or drag.  
-Similarly, the canards you have on the front could potentially go further forward, which would give them better leverage.
-I believe those Mk3 engines mounts have a reputation for being draggy, and a bit heavy.  And I know those 1.25 bicouplers are quite draggy.  There lots of potential ways to set up the engines, but I was thinking maybe use the 2.5m quad-adapters behind some Mk 3 to 2.5m adapters) on the side nacelles, holding 4 Rapiers each, and a similar 2.5 m quad-adapter on the very back (replacing the the 2.5 to 1.25m adapter fuel tank) for 4 nukes.  
-Those Mk 1 fuselages you have your landing gear attached to are probably adding drag without a significant benefit.  I would either get rid of them, and mount the landing gear on the wings, or put engines at the back (e.g., if you need a place to stash 2 more Rapiers or nukes).
-That tilted cockpit at the front looks awesome, but is adding more drag than if you had it inline, i.e. at the very front behind your nosecone.
-The shielded docking port is a relatively draggy nosecone. If you don't need to dock with it, or need the extreme temp tolerance, you could replace with an NCS adapter + small nose cone, a shock cone, or a pointy fairing that you never deploy.  Those can all do at least 2400 degrees, I believe.  
Making all those changes, I tried taking off with 10 Rapiers (all amidships) and 4 nukes at the back.  I did not quite have enough power, but it was close, and I think I added some extra rocket fuel with additional adapter tanks.  Hopefully you should need less LFO with a slightly lighter / mores streamlined design.  
 
EDIT: one more thing I forgot to mention: rotating the tips of your wings up a few degrees (aka wing incidence) can save you a little drag in flight, and make it easier to take off.  
All that said, you're probably not going to see a dramatic boost in TWR or delta-v.  Your design is already pretty darn efficient; it's just that an Eeloo and back mission with no refueling is so far that it's pressing the mathematical limits of what you can do with these engines.  As mentioned, ions or fancy gravity assist maneuvers could help, but this looks like a real toughie.  
 

Is your ship sitting on the drills,  or on landing legs?  Looks like your legs are not extending.   Sometimes the drills will push your ship up when deployed (only the bottom portion can go below ground level),  and make it move around a little as they drill.  Maybe try extending the legs, or mounting the drills higher.

There are two ways to look at this one; purely mathematically and from a practical KSP design perspective.  
First, the math - you may have seen info on here about the Tsiolvovksy Rocket Equation.  If not you'll definitely want to check it out if you're interested in these kind of issues.   Basically, a rocket's delta-v (cumulative ability to change speed, and hence, go from one orbit to another) is a function of wet mass (fuel), dry mass (everything that's not fuel) and specific impulse (engine efficiency).  Staging is also a factor, since it allows a rocket to ditch dry mass once it's no longer needed, and to move to higher specific impulse engines.  
So, assuming hypothetically that you can scale everything equally, if you want to double the payload mass but keep the same delta-v, you'd end up doubling the total mass of the rocket.  So there is no inherent greater or lesser efficiency from splitting a payload.  It should be fairly easy to picture this - imagine one rocket with half your total payload.  You can send two of these up by themselves, or you could staple two of those rockets together, but they'd perform essentially the same. 
 
Now from a practical perspective, you're not really holding everything else equal when you scale up a rocket to increase a payload.  You're probably using different engines, you're staging differently, and so forth.  In general, these differences mean that it's generally more efficient to send up one big payload.  A few reasons:
In KSP, engines generally get more efficient (both in terms of thrust-to-weight ratio and specific impulse) as they get bigger.  So using one big engine will likely end up better than several smaller engines.   You don't have to double up on certain one-per-rocket parts like probe cores, so you may get some mass savings here.  If you can stack your two payloads vertically, this will probably improve your aerodynamic performance and reduce drag.  Basically, the more mass you have versus a given surface area, the better your rocket will cut through the atmosphere (the technical term to look at is "ballistic coeffificent").  Note that. conversely, this also makes bigger stuff harder to deborbit or aerobrake.   Finally, you won't have to pack whatever hardware you were going to need to dock your various payloads.   
All that said, there are a few cases when splitting up payloads may be better.  If your full craft is really un-aerodynamic (think something like the International Space Station), you may be better splitting it up into different pieces.  Another reason might be if your bigger rocket hits some really awkward breakpoints, like it's too heavy for one Mammoth but two is overkill.  But you can generally design around this issue by adding or subtracting fuel so suit each stage's thrust.   

There is an option in the menu for "fuel transfer obeys crossfeed rules" or something like that.   Try turning it off.   I seems to have problems once in a while. 

Pretty sure any antenna is fine.  But you can check by launching it and, on the pad, checking the little contract pop-up window to see if that element of the contract is green.  If not, recover our revert and change out the part.

Does your path show a good altitude over the Mun before the SOI transfer?  Are you warping through the transfer?  That's known to lead to bigger errors when the same recalculates the new SOI.  
In any event, you can fix that orbit pretty cheaply by doing a "radial out" burn until your periapsis reaches the desired altitude.  It will be cheapest if you do it as soon as you change SOI.

I think so... All it really cares about is that you land and recover the part on Kerbin, not how you get it there.
But I have never tried attaching a target part for a contract like this, so can't guarantee that KAS would not override the data the contract uses to identity the part as the target.
You can always use the alt-f12 menu to complete the contract if some bug prevents it from happening automatically.

When I've had this problem, it's because the parachutes were accidentally being deployed in an earlier stage.  If so, you can right click them and select "disarm" or something like that.  Then redeploy after it's safe.  You might have to right click again to redeploy; can't remember if the staging option comes back.