From what I've read, most Varmint hunters like .243 and smaller caliber. The most popular seems to be the rem .223, just because it's such a common and inexpensive round.

Lots of guys prefer the .22-250 because it's flatter shooting. The .204 looks like a cool caliber, and there would be less pelt damage -- but it's new. Sometimes, it's good to let other guys buy the new caliber, and just wait and see whether it works out well or not. The .17 Rem is used too, but it's a bit more rare than some guys want to fiddle with. The .17 HMR is good four small game, but a bit too light for coyotes.

The .243 is popular because it can be used on white tail, pronghorn, etc. and it takes down a coyote easily. I think that round would tear up the pelt a bit though.

If all you want to do is eliminate nuisance animals, use whatever rifle you like to shoot. You can load a .30-06 round to take out coyotes if you want. (Prolly not to good for crows though)