AHHans

Well, you don't need an extra antenna if you have a relay (network) in place, that the "communotron 16"-class antenna in the control station can reach. The extra deployable antenna "just" always allows you direct connections to the (fully upgraded) KSC if you have line-of-sight.
I believe that the deployed science uses the same path finding algorithm as all other craft, so it may or may not use a certain relay depending on what gives it the best connection.
But do I understand you correctly that you didn't have a working connection all or most of the time until you got the rover into place? Did you get several "experiment generated science but is lacking a connection" messages?
Yes, that's IMHO a bug in the implementation.
I think that the deployed science works in the way that it checks every now and then (in the first version way too often, now (in 1.8.1) maybe every 10 precentage-points of generated science) if it is powered, there is a connection, and it has science to transmit. Then you either get a "experiment generated and transmitted X amount of science" message, or a "experiment generated science but is lacking a connection" message (or a "experiment is not powered" message, but that shouldn't happen with RTGs). If it doesn't have a connection at one time but does have one at a later time, then it will transmit all the accumulated science. But if it already is at 100% generated science then it will not check again, even if not all (or none...) of the science has been transmitted.
The only reliable way to fix that is to walk up with a Kerbal (preferably a scientist) to the experiment, pick it up, and place it again, thus resetting the whole experiment. I don't know if "just" walking up to it, and hitting "enable experiment" in the PAW will work.

Well, if you look in the save-file for the entry of said Kerbals, then there is is a section called "CAREER_LOG" (and "FLIGHT_LOG", but I think for dead Kerbals that's empty) that lists all career related events of the Kerbals. That should at least tell you on what kind of mission they were (which CBs they visited) and in which SOI they were for their fatal mission.
E.g. if the last entry before their death was "Suborbit,Kerbin" then they probably had a fatal accident during re-entry.

Depends on what you want to do.
Nuclear engines are heavy, next to useless (well, not that bad, but sill bad) in an atmosphere, have on engine gimbal, but have very good vacuum efficiency and decent thrust. They also only burn liquid-fuel and will leave any oxidizer untouched.
Spark engines are lightweight, can gimbal, and have decent efficiency in atmosphere and vacuum.
So the unexpected readout is because you probably have the delta-Vee displayed for being at sea-level on Kerbin (this is the default) and because you are using the rocket-fuel tanks which contain both Lf and Ox - so the nuclear engines have less than half the fuel mass that they will use with the left-over mass adding to the dry weight of the craft. If you click on the "delta-V" tab on the bottom right-panel, select to display all parameters, set the environment to "vacuum", and then "show all" (stages that is), then you can see that: the ISP (roughly translated the "rocket efficiency") for the nuclear version is much higher, that it's TWR (thrust to weight ratio = how fast the rocket accelerates) is higher, but also that its start mass is higher (because the engines are heavier), and that the end mass is much higher than on the spark version.
So nuclear engines are great for transfer stages, that only need to work in vacuum and where high dV is important (when fed from the right kind of fuel tank). Spark engines are good for small, lightweight craft (probes, landers) where their small size and low weight is important and the low thrust and lower efficiency (than e.g. the terrier engine) doesn't matter.
You probably have an idea how to go on from here, if not then please ask.

Are you asking how to launch into a polar orbit, or how to use the cheat menu to get into a polar orbit?
If you want to launch yourself, then - as @Vanamonde said - just launch due north or due south. If your goal is to use the M700 Survey Scanner then I don't think you'll need to correct for the eastward momentum you get from Kerbin's rotation, the scanner isn't so finicky.
If you want to cheat, then set the "Inclination" value in the cheat menu to 90 degrees.

Kind of what I thought. Is the craft designated as debris or an asteroid (in the craft naming dialog)? (I don't really think so, because IIRC that means that you don't have control of the craft at all when you get back to a craft that is designated like that.) So my best guess is that there is something wrong with your KSP installation. Either some problem with one or more of the mods that you are using, or your installation of KSP itself is corrupted.
Hmmm... does the probe use parts from mods or is it stock? If it is stock, then can you put the savefile somewhere where I could have a look at it?
Edit: O.K. our postings overlapped. Yes, sharing the savefile might help.

How much science you get out of an experiment doesn't depend on the quality of the radio connection at all. Even the weakest connection will give you the full transmittable science if it is good enough to connect at all - and your electricity doesn't run out, but that's another topic. Some experiments (e.g. crew- and eva- reports) will give 100% science when transmitted, others (e.g. surface samples, but also mystery goo observations and science jr. materials studies) give only a small fraction of the total science when transmitted and not recovered.
In your first screenshot you see that if you recover the mystery goo observation you get 40 science, and if you transmit it you'll get 16.8 science. That is a property of the mystery goo observation experiment, and will not improve with a better radio connection.

My big asteroid tug uses five Klaws at the end of large pistons to attach to asteroids. That allows me to adjust the length of the pistons to orient my craft to point exactly at the center of mass of the asteroid, and then lock the pistons for a solid connection. Sometimes one of the Klaws touches the asteroid at too steep an angle to get hold, but the even with only four Klaws connected the connection is usually good enough to get a solid grip and push around even the heaviest asteroids.
[Grrr... Google doesn't like me removing their cookie from my browser each session, so they decided to liquid me off. So right now it's too much hassle to log into my (an) imgur account to show you a picture.]
Edit: I finally bothered to try imgur again, so here is a picture of my craft (but also check out @PTNLemay's solution):

You don't actually have to have same vessel interaction enabled to have two docking ports (junjors only?) on the same vessel dock to each other.
I just did a test craft, and can confirm that (with juniors only!). Two docking ports on the same vessel can be docked by moving them together on hinge (or another robotic part), but cannot be undocked via an action group. Have them on two different vessels, and then the undocking via action group works.
You should check first if there is already an issue about this, but otherwise: yes.

Which version of KSP and BG are you running? That was a common bug in the first release of BG, but it was (mostly?) fixed in one of the KSP 1.7.x updates.
You can go there with a Kerbal and switch it back on again.

You should be able to get the craft file with my changes here: https://drive.google.com/open?id=1KRTN5QhF7Xt5brd4ki0R1TuuOPRbFeEi Have a look at the settings for the helicopter blades and the up-down axis action action group to see what I changed.
To take off you first need to start the engines (press <1>) and set engine RPM to max (press <Z>), then you can use the up-/down-translation controls to change the blade pitch: <K> will increase blade pitch and thus increase the lift, <I> will decrease the blade pitch and lift. I usually keep the window of the one the helicopter blades pinned in a corner of the screen, so that I can see what the blade pitch settings are (I.e. the value for "Deplay Angle" in the window).

Well, yes, that orbit won't work for Duna! (And also not for Dres because the inclination is wrong.) You could try to see if an orbit that is just inside Duna's orbit will be accepted as being close enough to the contract orbit. But, to be honest, that doesn't look likely to me it's just too far out. (Or did you try that already?)
In such a case I would use the cheat menu (on Linux <Right-Shift> - <F12>, on Windows <Alt> - -<F12> or so) to mark the contract as completed. In my books this is working around a bug and not cheating.
Are you using any mods that could mess with the contract generation system? Or is this a bug in the stock game?

Meh!
I had the problem with my first spaceplane designs that they flew fine at first, but when I switched to rocket propulsion and pitched up (ca. 10 deg, so significant but not excessive) then they flipped out of control. Took me a while to figure out the problem.
Well, I'd say that "best practice" is what @bewing said: try it out. Shifting around fuel to adjust aircraft trim is definitely not a crime but good practice. (Even in RL.)
That's about it. Maybe together with "just because it's stable low & slow, doesn't mean it has to be stable high & fast". (Which - as I said - took me a while to figure out.)
I don't really know. I guess it is essentially in the geometrical center, but there are some parts around with a surprisingly high drag, so that can be off.

Well, to me it looks like you have the control point pointing upwards not forwards. (And, yes, in this case the wheel control will work mostly fine.) It could be that you have the docking port on the top of the rover set as the control point. Do you have the RoveMate as the probe core? (It is hard to see, but it looks to me like that.) make sure that you "control from here" on the RoveMate, and have it set to "forward".  (When you turn the rover left and right, the coordinate grid on the navball should move left and right and not rotate around. Once you have that, make sure that navigation to a target is activated and turn the rover around until you see the target marker on your navball. Drive in that direction - the prograde marker should show up on top of the target marker -  as the terrain and you rover allows, and you should get there.
You probably already noticed that driving a rover on Minmus is not pure fun. The low gravity and bouncy wheel suspension make for an unstable ride.