My buddy and I completed the game on hardcover mode using online co-op and when the game was finished, he got all achievements and I didn't get any for beating the game. Also at one point during the game, after a level, it said I had 0 kills, which was false. Is there any way to rectify these errors? Especially with achievements? 

 Well, in regards to the achievement, I'm not 100% sure but if it was his lobby they might be the one to receive the achievements for it. Again I'm not exactly sure, just a possible option.

 hi, my mate and I were doing quick run through on cadet before doing authentic and on the last mission he got mission 8 and completing cadet. i got neither.

even though my records shows i'd done it and got the collectables. i had to repay the last level and the two achievements popped.

so if you know the one where you got no kills it may be worth replaying that mission