jimmymcgoochie

When the rocket separates, what direction does the navball face when you control it? If it’s attached via a radial decoupler then there’s a good chance that that radial decoupler becomes the root part facing the wrong way; SAS tries to point prograde but ends up facing the completely wrong way and it spins out of control. Setting the probe core on the rocket with an action group for “control from here”  would probably fix that.
Alternatively, the weight of the engine and fuel at the back could be making it unstable and at a relatively low speed the little fins might not be enough to counteract its tendency to flip heavy end first. Bigger fins or a longer rocket might solve it in that case.

The bad news is, your save file is corrupted.
The good news is, KSP has backups! Go into KSP/saves/<your save name>/backups, find the newest persistent_[date+timestamp] file and copy it back to saves/<your save name> then rename it to persistent.sfs and overwrite the existing file. You'll probably lose a little bit of progress but it's better than losing the entire save. You can also rename a named save file to persistent.exe but depending on how regularly you save you could lose a lot more progress that way.

If you go into KSP/saves/New Horizons/backups, you’ll find some backup save files which are date+time stamped. Grab the newest persistent-(time stamp).sfs, rename it to persistent.sfs and move it to KSP/saves/New Horizons, overwriting the existing one, then load the save in the game.

Right click KSP in Steam > Properties > Betas, you should get a list of all the versions of KSP, pick the one you want and it'll automatically download.

You put mods into the Steam copy of KSP. Big no-no, Steam breaks modded KSP all the time and usually in weird ways like this.
Copy your save files to your desktop. Uninstall all your mods. Uninstall and reinstall KSP through Steam. Verify the game files in Steam and run the game to make sure stock KSP works. Copy KSP in its entirety from Steam/steamapps/common and paste it somewhere outside of Steam’s folders so it can’t interfere. Install your mods into this new copy and add your save files to it.

Let's take a step back from this OpenGL stuff and just focus on the immediate problem- broken mods. 
First of all, don't put mods in the Steam copy of KSP. Just don't. Steam breaks modded KSP, especially if you're switching versions with mods installed.
There's a good chance that reinstalling KSP and then copying it to use mods will solve your problem, so try this:
1.       Right click KSP in Steam library > browse local files.
2.       Copy KSP/saves/<your save name> and paste that on the desktop.
3.       If you used CKAN to install your mods (and you should!), click File > export modpack and save that to your desktop too. Screenshot your GameData folder with mods installed (even if you use CKAN) so you can check that you’ve reinstalled them all later.
4.       Uninstall all mods from KSP.
5.       Right click KSP in Steam library > properties, disable Steam cloud.
6.       Completely uninstall KSP through Steam.
7.       Reinstall KSP through Steam, right click > Properties > local files > verify integrity of game files. If you want to play a version other than the current release (1.12.3), pick that version in the Betas tab and verify the files once Steam has installed that version.
8.       Run KSP and make sure it loads properly without mods.
9.       Browse local files again, then go up one level (to Steam/steamapps/common) and copy the Kerbal Space Program directory, then paste it where you want to keep it- make sure it’s outside Steam’s folders so it can’t meddle with it in future.
10.   Rename the KSP folder so you know what mods you’re using in it (e.g. 1.10.1 RO/RP-1, 1.12.3 JNSQ), then add this new copy to CKAN and use the modpack created in step 3 to reinstall all your mods; or reinstall them by hand if you don’t use CKAN. Double-check that all files and folders you had in GameData before uninstalling/reinstalling everything are there again, if not then you’re probably missing some mods.
11.   Move your saves from your desktop into your new KSP copy’s saves folder, run KSP, load save. A warning about vessels having missing parts is usually because a mod is missing or wasn’t installed correctly.
If that still doesn't work and the game won't load, we'll need the log files along with a full mod list, a screenshot of your KSP GameData folder and a screenshot of where the game loads to before freezing. If you don't know how to find the logs, this guide will tell you where to look:
 
Put the log files on a file sharing site and post a link here, please don't try to paste the entire log files directly into the forums as it's nearly impossible to read them like that.

The game doesn’t check power on the relay, only the vessel transmitting. Relays will work regardless of whether the probe core is active or even if it has no power at all.

Gravity assists are your only option. Plonk down a quicksave first, then set up a node as close as possible to your vessel and tweak the trajectory so you get an encounter with Tylo; use the fine adjust tool in the bottom left corner to fine-tune the encounter so that Tylo’s gravity slows you down relative to Jool, possibly even into orbit, though with a fairly high-energy transfer like that it’s unlikely to be enough on its own. A retrograde burn at Tylo periapsis will be more effective than doing it around Jool and all you really need to do is capture into any orbit so you can come back down for future gravity assists from Tylo and/or Laythe to bring your apoapsis down so that Laythe capture is cheaper.
Done properly, a gravity assist can enable a Jool capture and a visit to any and all of the inner three moons without using any fuel at all:

 
(This is a slightly altered Jool system but the three main moons are all in the same places)

Gravity assists are your only option. Plonk down a quicksave first, then set up a node as close as possible to your vessel and tweak the trajectory so you get an encounter with Tylo; use the fine adjust tool in the bottom left corner to fine-tune the encounter so that Tylo’s gravity slows you down relative to Jool, possibly even into orbit, though with a fairly high-energy transfer like that it’s unlikely to be enough on its own. A retrograde burn at Tylo periapsis will be more effective than doing it around Jool and all you really need to do is capture into any orbit so you can come back down for future gravity assists from Tylo and/or Laythe to bring your apoapsis down so that Laythe capture is cheaper.
Done properly, a gravity assist can enable a Jool capture and a visit to any and all of the inner three moons without using any fuel at all:

 
(This is a slightly altered Jool system but the three main moons are all in the same places)

It might be USI and/or MKS which can recycle existing parts into usable resources to make new ones; alternatively use the explosives in KAS to destroy parts completely.

How long is a piece of string?
In the real world, fighter jets can range from something like the English Electric Lightning, which was little more than two jet engines with wings and a cockpit stuck to them with a minimal flight load to make it a rapid response fast interceptor, all the way to a heavy air superiority fighter like the F-15 Eagle which can carry three external drop tanks and (if my memory is correct) can fly around 5000 miles across the Atlantic without stopping, relying on sophisticated electronic systems and guided missiles rather than close-quarters agility.
How much fuel you need depends entirely on how far you want to fly and what kind of plane you’re building- a smaller, more agile plane will need less fuel to give it that agility but this will cost it flight range, whereas a plane built for greater endurance will suffer from the extra weight of fuel it has to carry around with it.
Cruising in “dry” mode will save you a lot of fuel when you don’t need the extra thrust of the afterburners.

How long is a piece of string?
In the real world, fighter jets can range from something like the English Electric Lightning, which was little more than two jet engines with wings and a cockpit stuck to them with a minimal flight load to make it a rapid response fast interceptor, all the way to a heavy air superiority fighter like the F-15 Eagle which can carry three external drop tanks and (if my memory is correct) can fly around 5000 miles across the Atlantic without stopping, relying on sophisticated electronic systems and guided missiles rather than close-quarters agility.
How much fuel you need depends entirely on how far you want to fly and what kind of plane you’re building- a smaller, more agile plane will need less fuel to give it that agility but this will cost it flight range, whereas a plane built for greater endurance will suffer from the extra weight of fuel it has to carry around with it.
Cruising in “dry” mode will save you a lot of fuel when you don’t need the extra thrust of the afterburners.

Sounds like an exception in the loading process is causing the game to stop loading, however the loading screens and tips are separate to that so they'll just keep on going even when the game will never load. If you got KSP via Steam, verify the game files (right click KSP in Steam > properties > local files) and if that doesn't fix it or you didn't get KSP via Steam, back up your save games and then uninstall and reinstall KSP.

I suspect that’s a mistake in the contract text, it should probably be “have an engineer on the station”.

Try recovering the cursed plane from the Tracking Station first?

KSP/saves/your-save-name-here/ships/subassemblies I believe?

I believe it’s Magpie Mods that makes Restock/+ parts shiny, along with a variety of other mods’ parts, and I think it uses TUFX to do so so you’ll need both.

TUFX and KS3P might do what you want- both are post-processing mods that change how the game looks, which could include the background blurring in that loading screen.
If you also want the additional KSC buildings featured in the loading screen, you’ll need KSC Extended- this has been updated since that loading screen was made so it doesn’t look exactly the same, the diagonal road coming off the crawlerway isn’t there any more. It also adds some extra launchpads which are functional (you can launch rockets off them!) and replicate some of the real launchpads at Cape Canaveral/Kennedy Space Centre.

Back up your saves and then uninstall and reinstall the game?

A few things stand out to me:
The logs seem to indicate that you're missing ROLib, a dependency of various RO mods. How did you install these mods and when?
The logs also seem to be incomplete, I'd expect to see something at the top with the KSP version and then either your system's RAM or VRAM (GPU) depending on which log it is. Which version of KSP are you using?
You've installed mods into the Steam copy of KSP. Don't do that, Steam breaks modded KSP a lot- instead, create a copy of KSP and install your mods into that:
1.       Right click KSP in Steam library > browse local files.
2.       Copy KSP/saves/<your save name> and paste that on the desktop.
3.       If you used CKAN to install your mods (and you should!), click File > export modpack and save that to your desktop too. Screenshot your GameData folder with mods installed (even if you use CKAN) so you can check that you’ve reinstalled them all later.
4.       Uninstall all mods from KSP.
5.       Right click KSP in Steam library > properties, disable Steam cloud.
6.       Completely uninstall KSP through Steam.
7.       Reinstall KSP through Steam, right click > Properties > local files > verify integrity of game files. If you want to play a version other than the current release (1.12.3), pick that version in the Betas tab and verify the files once Steam has installed that version.
8.       Run KSP and make sure it loads properly without mods.
9.       Browse local files again, then go up one level (to Steam/steamapps/common) and copy the Kerbal Space Program directory, then paste it where you want to keep it- make sure it’s outside Steam’s folders so it can’t meddle with it in future.
10.   Rename the KSP folder so you know what mods you’re using in it (e.g. 1.10.1 RO/RP-1, 1.12.3 JNSQ), then add this new copy to CKAN and use the modpack created in step 3 to reinstall all your mods; or reinstall them by hand if you don’t use CKAN. Double-check that all files and folders you had in GameData before uninstalling/reinstalling everything are there again, if not then you’re probably missing some mods.
11.   Move your saves from your desktop into your new KSP copy’s saves folder, run KSP, load save. A warning about vessels having missing parts is usually because a mod is missing or wasn’t installed correctly.
 

KSP 1.8 had a big Unity update and any mods that weren't recompiled against it will not work in KSP 1.8 or later. Either delete all mods not marked as compatible with 1.8 or later or go back to KSP 1.7.3 to use them.
There's a more up to date version of Kopernicus Expanded available here:
 

Did you turn the CommNet display off? There’s a button beside the signal indicators that can toggle what lines are displayed- off, first hop (the thing you’re directly connected to), path (all connections back to a ground station), vessel links (everything the vessel can connect to directly) and network (everything) and the same button can also be found in the tracking station.
Do you have CommNet switched on in the difficulty settings? If it’s turned off then you’ll always have a connection but won’t see any signal lines.

I think if you leave it running in the background, it'll do that automatically. Worth a try at least, if it doesn't work you can always load an earlier save.

Like this:

*boop*

Like this:

*boop*