Sorry for the delay, it's been quite hectic and I couldn't think of anything for a while (regarding this).
The crash to desktop (sometimes not completely, in that case try alt+tabbing untill you see the BC logo, assuming it's there, and when you find it, press Alt+PrintScreen to make a screenshot and attach the screenshot to this thread) after you exit Quick Battle (or the game) is a "known" problem, but no real solution is out there as of yet.
The crash during the shakedown cruise is mostlikely caused by a missmatch of torpedoes. When the Bird of Preys enter the set the scripts reload the quantums, or tries to anyway, but this can crash BC if the ship you are using doesn't actually have the quantums.
To fix this, you first need to open up scripts/Ships/Sovereign.py in notepad, and look for: "
"HardpointFile": "<<Something>>",
Then, go to scripts/Ships/Hardpoints/ and look for <<Something>>.py and attach it here, I'll make sure it doesn't crash BC (assuming it's this).
As for the save points, in case of the shakedown, I believe there is a savepoint right about there. But generally, with mods installed, the savegame system doesn't work correctly (some people have got it working, but it largely depends on quite a number of factors), so you may have to continue your single player from the Custom Missions menu.