I know the kills don't count towards something like a kill streak but do they apply to achievements (maybe contracts?) as well as total score at the end of the game?
The kills and calling it in counts toward some contracts.
The kills are also worth points themselves (and calling in the kill streak itself gives you some points) so they do effect your score.
The kill streak helps out the following challenges:
(I may have missed a couple since I just skimmed the challenge list)
It may also help with a few other challenges such as blowing up cars, shooting through things, defending points, etc. but not directly.
As Bora mentioned in the comments you also receive more points for head shots done as the chopper gunner.
It looks like the kills count towards contracts (Depending on the contract of course) and your final score.
So when someone has a final K/D of like 40 and 5, he may have gotten into the chopper gunner.