It depends on the animal. I'm not sure how I classify them exactly but I do... Somehow.
Deer, kill to eat or give to those who will.
Crows, they're an infestation. Kill every one I can, which is not many since I don't try.
Starlings/Pidgeons, kill em all, let God sort em out.
Woodchucks, kill them all. Don't generally swerve to hit them but don't try to miss either.
Opossum/skunk/rabbit/coon, never really bothered any of them. Except a few rabbits back when.
When I was a kid with a BB gun I would shoot birds all day long, partly I think because it started with genuine pests (starlings and pidgeons) and my grandparents farm. The "nonpest" birds my dad was always like "why you shoot the birds? Leave them alone."
Funny, now I wouldn't kill the nonpest birds for anything.
As for running stuff over, the only ones I don't avoid are woodchucks.