If the rifle is accurate, then you shouldn't have any problems killing rabbits out to 25 yards or so. A headshot is the best, but a broadside or quartering away shot to the vitals will also do the job nicely. My brother and I used to kill rabbits all the time with a fairly inexpensive crossman pellet rifle when we were growing up (garden pest control was one of our chores). We liked to use the pointed pellets because they seem to improve penetration quite a bit.

I think that a raccoon is too big for all but the top-end, ultra-high velocity (relatively speaking) air rifles. And a coyote is just way too big for any air rifle. You'd be very likely to just lodge a pellet into a muscle or against a bone, likely resulting in infection and a slow painful death for the animal. I know people use .17HMR and .22WMR for coyote at short ranges, but I personally wouldn't take a shot at a coyote with less than a .22Hornet.

Just out of curiousity, why not a .22LR? It'll take small game up to raccoon in size, and is dirt cheap to shoot. .22LR rifles are usually inexpensive too. A .22LR can be loaded with .22CB Longs for short range use if noise is an issue.

Mike