RE: 243 for bear
Bear? Don't know, but I wouldn't try it.
Coyote? Depends on what your intentions are. If you just want to control their population and don't care about the pelts, then the .243 with 75-80gr varmint bullets would be very effective. If you want to skin them and sell the pelts, however, the .243 might do a little more pelt damage then you might like. In that case I'd say try a .223 or 22-250.
Mike