Echo__3

You need to have the engineer get very close to the satellite. Then attach the battery using the construction menu. There is a crane hook on right side of your screen for this.

I mean a probe used as a control point if you don't have a Pilot.
It sounds like you decouple the rover from the station and remain in control of the station. All you have to do is press a bracket key [ or ] and switch over to the rover.

Three relays around every planet and moon should be sufficient. Check out the wiki to help calculate what antennas you'll need. CommNet - Kerbal Space Program Wiki.
Here are some other helpful links.
 
https://ksp-visual-calculator.blaarkies.com/
https://wiki.kerbalspaceprogram.com/wiki/Tutorial:Ideal_Orbits_for_Communication_Satellites
https://wiki.kerbalspaceprogram.com/wiki/Tutorial:Setting_up_a_CommNet_system

You cancelled the flow of liquid fuel from the tank. Look at the readout for the Rockomax X200-16 Fuel Tank and click on the red strike out icon.

I found I could reliably get to 15 km with my propeller designs. Anything higher I considered a bonus. At 15km the atmosphere on Eve is the same density as sea level on Kerbin, so more engines become viable options. I made this design, and it worked pretty well. Probably should have put a couple parachutes higher up though.
 

Yes, you can use the map backwards to calculate your return back to Kerbin. So the landing figure is equal to the return to orbit. Make sure to budget extra. The maps usually reference an almost perfect suicide burn landing.

It has just a little less gravity than the Mun, so a Mun lander will work just fine. It does take a fair amount of ∆v to get there and back.  You already know how to get there, but you may need 2900 m/s of ∆v to get back to Kerbin.

The vector engines are quite heavy. It is quite likely that your center of mass has shifted too far to the rear of the plane causing it to be unstable. You could try moving the main wings back a meter or two and see if that helps. You can also place docking port angled 20 to 40 degrees in the cargo bay and use the control from here option. Now you can have SAS hold prograde and keep your craft at a higher angle of attack.
 

You are already orbiting Gilly, so you cannot nor do you need to select it as your target. In the game, you cannot target an object that you are orbiting. Notice too that you cannot target Eve or the Sun as you are technically orbiting them as well.