It does cost them just to process the claim because the other guys side is unlikely to completely roll over if there is any chance to reduce the payout. Obviously its not as much as the claims themselves but it also isn't free.

Yep. I have a friend that was plowed into from behind while waiting at a red light, twice within a few months time. Two separate intersections. Totaled her car both times. Her insurance rates went up, even though she was clearly not at fault.