FullMetalMachinist

In 1.2.1 the default was changed from Backspace to Back Quote ( the ~ button) to avoid clashing with the default 'Abort' action group.
http://wiki.kerbalspaceprogram.com/wiki/1.2.1
 

If you turn that on then the legs will ignore being shielded by a cargo bay or fairings. So no more 'cannot deploy while stowed' business. 

Yes, but that's only half of the story. The formula for finding range between two antennas is: sqrt(antenna1 * antenna2). 
Not quite, see the formula above. 
Regular antenna will work on the controlled ship. 
The range does increase. That (combinable) tag after the range means that you can combine multiple for more range. Unfortunately, I forget what the formula for combining antennas is, but I'm pretty sure it's not linear (i.e. 2 of the same antenna don't get you twice the range). 

Yep, it's one of the new Advanced Tweakables. Turn them on by going to Main Menu -> Settings -> General -> Advanced Tweakables. It's right above the new slider for Default Pre-launch Throttle. 

There has not been any update or patch to the released game since v1.1.3 came out in June. So that leave three possibilities for your troubles. 
1) You installed a mod that changed something, whether it was supposed to or not. I think you said that you're running stock, so probably not this, but wanted to be sure and mention it. 
2) You're doing something different, whether with your craft design or your piloting. Really only you can answer that. I would suggest loading a ship that you've flown successfully before, and fly it using the same flight path, and see if it still works. 
3) You've encountered some bug or some corruption with your game. First thing I would try is to go into your main KSP directory, and delete the file called "physics.cfg". This will force KSP to create a new file using the default values the next time you run the game. If that doesn't work, try a clean re-install. 
Hope this helps, and if it ends up being the craft design/piloting that's the issue, then screenshots definitely help to diagnose problems. 

Since you already have KER, this is pretty easy. 
You can customize the different info panels and add more things to them. So edit the "orbital" one to include the "time to AN/DN." Then when that time gets close to zero, you're pretty much right on the node. Then you make your normal/anti-normal burn to zero your inclination. 

It's a bit long, but this article is what made me really understand how gravity assists work, and how to use them in the game. 
The TL;DR of it is that if your orbit after the assist looks more similar to the assisting body than it was before, then that will speed you up. If your orbit after the assist looks less similar than it was before, then it will slow you down. 

Known bug. When any of the contract orbit parameters are either N/A or NaN the orbit lines will not be drawn. 
http://bugs.kerbalspaceprogram.com/issues/9651

Yes, that was an intentional change to the way labs work. It was meant to prevent you from taking the same science result to several labs and getting unlimited science. 

Sounds like you're having the same issue as the person in this thread. 
Basically, one of the orbital parameters is "N/A" for some reason, and because of that the game can't draw the orbit. 
It's certainly a bug, and you can either use the debug menu and force the contract to complete, or just use the numbers given to know where to put your satellite. I think as long as you get the inclination, PE and AP correct, it'll work. 

This was a deliberate change in 1.1.x.
There was a small handful of things that you used to be able to do with probes with no power that you shouldn't have (extending landing gear was one, I think). Part of the fix for those included not being able to turn on a battery that was disabled.
It makes sense if you think about it. If there's absolutely no power for the probe, how does it know, or be able to, turn them back on? 
I know that previously this was a nice "safety feature" to leave one battery turned off, and if you forgot to extend solar panels you could turn the reserve battery on and get your panels extended. But, alas, no more. 
Now, I see that your ship has a crew pod on it. So if you EVA a kerbal over, they can board and turn the power on, but just the probe core cannot. 

No transmission penalty, because you're sending science points and not science data. The new thing in the changelog that you mentioned just means that if a lab has researched points, you have the option of recovering instead of transmitting. Previously if you recovered a lab that had researched science points in it, you wouldn't get those when you recovered. 

Yeah, you're trying using it. 
Patcher has been broken for ages. You have to re-download the whole game again from wherever you bought it (Steam being an exception, it'll download the update for you). 

With just giving it a quick look, I see two things that you could change. 
First is to move the rear landing gear forward towards the CoM. The gear act like a fulcrum that the plane has to pivot around in order to take off. The CoM is the natural rotation point of the plane, so the closer those are together the easier it'll rotate for takeoff. 
Second is that you can add some angle of incidence to the wings. This is where, if looking at the screenshot, the wings go up and to the right, so that the front edge of the wings is higher than the back. The reason to do this is because in KSP wings will only generate lift if they have some angle of attack. Adding some incidence gives them a small angle of attack while the rest of the plane is at 0, and lets your plane lift off without having to pitch up very much (or at all). 

I know you specifically said that you don't want to search around for this and would rather someone simply explain it to you here. Sadly, I'm not able to do that for you. 
I can, however, link you to a really good thread by @Claw, where he wrote an extremely good and easy to understand breakdown of what's going on and what that slider does. 
 

@ii_DyNaMiCs_ii may I point you towards the 1.1 section of the forums?
http://forum.kerbalspaceprogram.com/index.php?/forum/78-11-pre-release-branch/
Though, to be honest, I don't expect you to find a lot of help there. Nothing against you, it's just that's the nature of the pre-release, it's buggy as all get-out. 

Well, the only way to do that is to get rid of those parts . 
The other thing you can do is fly with very little angle of attack. The problem with that is that on reentry you need a high AoA in order to bleed off speed. 
Are you starting to see now why building and flying a successful shuttle replica is considered one of the hardest things to do in KSP? 
The answer to your current problem is a redesign that puts as much weight as possible up front, moving the CoM forward and giving your wings and control surfaces a similarly long lever arm to counter the drag forces. 
The question on exactly how you do that is a little tricky, primarily because you have those three big, heavy engines stuck to the back. I would start by putting any monoprop fuel tanks and anything else you can think of in between the cockpit and the cargo bay. Also you can add canards to the front, though that destroys any thoughts of "purity" with regards to the replica status of your shuttle. 
Also, try to move the main wings and control surfaces as far back as you can, and make sure you have a set of body flaps. (if you don't know what that is, do a Google Image search for "sts body flap") 

IIRC you have to have two <thing> in a given place before the contract system will offer you contracts based off them. 
For instance, you need two space stations around Kerbin before you'll be offered to do anything to them, or two bases on Minmus in your case. 
Don't take this as Gospel truth, but I'm pretty sure @Arsonide is the one I remember saying that. 

You can hold shift to get finer angle snap increments. This works both while placing the part and using WASDQE or when you're using the rotate gizmo. It also allows finer angle snap for the location when you're placing the part and when you use the translation gizmo. 
Unfortunately I don't recall exactly what angle increments it gives you. Maybe 5 degrees? 

Generally speaking the reentry angle has more to do with how much heat your ship can handle, and less to do with slowing your ship enough to deploy 'chutes.  If you're not burning up on reentry, then your angle is probably fine.  
However, slowing down enough for 'chutes is more controlled by your ship design than any flight path (assuming you don't burn up).  And for that to get sorted out, it'd really help to see a screenshot of your ship to see what the problem might be. In the mean time, I put together a little experiment going off of your (slightly vague) description of your ship and reentry profile. 
This first album is a fairly 'normal' early career tourism craft.  While it needs a little help keeping the right orientation during parts of reentry, it's pretty straightforward.  Note in the 4th and 5th pics how I'm using body lift to slow down my vertical decent.
Now this next album is the same exact ship (it's actually the same reentry, I loaded a quicksave), but I "accidentally" let control get away from me, and wound up pointing prograde. This is generally a bad thing for slowing down, but I was still able to (barely) deploy my 'chute at the last second. In the first picture, I'm still "pulling up" to give me the same slowing benefit of body lift, it's just so little that you can't see the cyan lift vectors (note the small AoA on the navball).
There are varying configurations of this basic design for early career craft. The ones you need to be really careful of are if you have two or more crew cabins stacked in line under the command pod. This makes it almost impossible to hold retrograde the whole time, and if you flip around makes it again almost impossible to slow enough for 'chutes.  In that case, I'd suggest something like this:
Upside down heat shield on top (with an expendable nose cone to help on the way up, it'll burn up on the way down) and radial 'chutes on the bottom. Notice how it's inherently stable while pointing prograde because it has the heavy heat shield/command pod combo on top, with the big, light crew pods on bottom (like an arrow).  This way it'll want to point the blunt end of the heat shield to the front, which will help tons with slowing you down.  
These are just some of the "general rules of thumb" for reentering crafts. For more specific advise, I'd suggest uploading a screenshot of the ship that's causing you problems.

I'm no expert, so I don't know if I can explain the "why's" very well, so I'll just show you instead.  I took your design and modified it slightly.  The main difference is that I added another fuel tank to give the body some more room to work with, and moved the deluxe winglets up some, and added control surfaces at the back (most notably a vertical stabilizer/control surface). It's very important that each control surface only has one job. So the elevators at the back control pitch, the winglets control roll, and the rudder controls only yaw.  
Also, I'm not sure if you're playing career or not, but I tried a version that uses an Engine Precooler instead of the radial intake. It has less drag but still provides enough intake air, so it's generally better. But if you're in career and don't have it unlocked yet, the radial intake works fine. 

I actually had a lot of fun with this little plane, it's really fun to fly. The only thing is those wheels are really short, so be careful you don't hit the engine when you take off or land.  You had a good start with the design, just needed a little tweaking.  
Here's the craft file if you want to look more closely at what I did.  Hope this helps, and happy flying!

My view on the question of balancing RCS thrusters is this: don't worry too much about it. There are a couple of reasons for that.  Fist is that for a ship that size, you really don't even need RCS for attitude control. The reaction wheels inside that command pod should be plenty to turn your ship.  That means that they're only useful when docking. 
Now, for docking there is a little-known trick for translation movements: use fine controls. If you're unaware, you can press Caps Lock at any time and it will put you into "fine" control mode, and the little red arrows in the bottom left corner will turn cyan/light blue.  The important thing this does for RCS and docking is to automatically adjust the thrust of each RCS block to keep it within what the command pod reaction wheels can counter, as long as they're fairly close to balanced to begin with.  This means that RCS thrusters closer to the CoM will thrust more, and ones further away will thrust less, automatically.  So as long as you're close enough to being balanced, the fine controls will take care of the rest for you.
Now, if/when you're willing to look at mods, there's a great one called RCS Build Aid that has lots of features to get your RCS balanced just so. But for most situations, it's really not necessary thanks to the handy "fine controls" trick.
As for your second question, very light ships (like small unmanned probes that you mentioned) are pretty terrible about wasting monoprop while trying to hold a certain orientation.  This comes from the PID controller for stock SAS being very twitchy and getting itself into a feedback loop where it tries to correct, then overcorrects, then tries to correct the overcorrection, etc., etc. My suggestion would again be to not rely on RCS for attitude control, even on reentry. (As a side note, the ship you have pictured will do fine on reentry, assuming that's a heat shield between the pod and decoupler. That configuration is extremely stable during reentry, so much so that you'd have a hard time getting it to point any direction but retrograde.)

My guess is that something with your persistent save file got messed up. If you have a quicksave (or a named quicksave from pressing Alt+F5) you can delete the persistent.sfs file (make a backup first just to be safe), copy the quicksave and rename the copy to persistent.sfs.  Then launch the game and you'll hopefully be able to load like normal.

Not at all true. If you're having problems with Steam, then don't use it. If you're willing to re-purchase the game directly from the KSP site, then feel free to do so. Just because the in-game patcher isn't working doesn't mean you can't get the latest versions of the game when they come out.  You just can't patch the existing game. When an update comes out, all you have to do is download that new version from the KSP site (sign in so it knows you've bought the game). Once you have the new version installed (which only consists of unzipping the download file), you can copy your saves and craft files over to the new game. Or just start fresh with the new update, up to you.
I understand your frustrations, and that it seems like a hopeless situation. But you really do have other options.

So here's the thing, and it looks like you might not know this yet.  KSP is amazing in it's ability to have lots of really cool features that make the game great, while at the same time not telling you about those features or how to use them. (Don't worry, the devs are currently working on fixing that for the 1.1 update.)
I think what you need is to simply adjust your staging sequence so that the parachutes are in their own stage. Do this by hovering over the staging list in either the VAB or while in flight. Tow buttons will pop up next to whatever stage you are hovering over, a + and a -.  The plus add a new stage at that spot, and the minus deletes a stage, but only if it's empty.  So go to the top of the list, press the plus to get a new, empty stage, and then click and drag the parachute icon into the top (empty) stage. Note that if there are more than one of anything in the staging list in symmetry (denoted by the small subscript number on the icon), you have to click once to open the group, then click and drag any one of the group to drag the whole group to another stage. 
If that's not the problem you're having, then a few screenshots would help bunches.  Ideally a few of them, one in the VAB, maybe one in space or on the launch pad, one during reentry when things get all explody on you.
Edit:  Forgot one thing.  The parachute icons can be a little tricky to interpret.  White icon in a grey background means that they are ready for staging, and that it is safe to do so. White icon in a red background means that they are ready for staging, but will be destroyed if you do.  White icon in a yellow background means they are ready for staging, and might be destroyed if you deploy them.  Cyan icon in a grey background means they have been armed (aka staged) but have not yet deployed (in this case they can be unarmed via the right click menu, placed in a new stage, and staged again later).  Yellow icon (I think, can't remember this one for sure) in a grey background means they are pre-deployed.  Green icon in a grey background means they are fully deployed.  Red icon in a (either grey or red background, again can't remember exactly) means that they have been destroyed by aerodynamic forces.  At that point, they can be repaired by an engineer of sufficient level, provided that you don't smash into the ground before that happens . Hope this helps, and if you have any other problems, let us know!
Edit 2: D'oh! How rude of me! Welcome to the forums @Giygus!