As jhm said, check your state's regulations to make sure a .22 caliber rifle is legal for deer. I know many states do not allow it. For double duty (deer/coyote) the .243 is about ideal, although the .260 Remington might be worth a look. Keep in mind that anything larger than a .223 is going to wreak havoc on the smaller predators - fox and bobcat primarily. If you're really interested in saving the hides, a .17 caliber or one of the smaller .22s (.22 hornet or .223) is the best option short of trapping the critters.
The .22-250 does move the bullet faster than the .223, but at practical calling ranges and responsible deer ranges, there is little difference. I would recommend, however, that you opt for something with a bit more punch for deer hunting.