1. Trapped animals die after a while, and presumably death comes faster if they were injured by the trap. It ought to be possible to build a containing "wall"* around an animal and then release it from the trap, though. Such catches have been reported to just disappear eventually (just as creatures can spawn in places where they wouldn't actually be able to path to).
2. Damage to any body part except the head will affect the carcass "quality". Thus, aim for the head and use blunt damage (to limit the damage if you miss the head). Large animals can take a fair bit of damage before the skin gets injured, while smaller ones can get a damaged skin by a single hit not on the head. I've noted that a shaft hit by a javelin usually doesn't damage the skin of birds, while a shaft hit by a spear often does, so the force of the impact probably affects the outcome as well.
*"wall": A fence for a large animal, and traps that can't target the type of animal for smaller ones in a trap line. Actual walls should work as well, but your prey will be dead by the time you're done if starting from scratch.