Question about the trainer - if you boost your stats on your guys, do they stick or do you have to do it every mission?
If you change the soldiers stats in the barracks they stick (just ensure that you save the game beforehand as the game has crashed a couple of times for me when messing around) Remember to save after you change soldier stats also. Nothing worse that loosing the new stats and maybe some progress if you didn't backup before the change.
If you change stats in the battlefield they do not stick so you will need to do it every time. (Remember to save!!!)
Question about the trainer - if you boost your stats on your guys, do they stick or do you have to do it every mission?
Depends, if you use the Barraks one its perm. If you do it in mission its for that one only (or thats how its always worked for me).