A .22 LR will do the job but most of the coyotes will run off wounded. Unless its a good vital shot.

You can get a coyote pelt all year round but the winter is by far the best since its real fluffy. The summer pelt is kinda simliar to a dogs coat.

30-30 will damage the fur. I'd go with a .20 caliber of some sort. Some good calibers are .223, .204, .243, and my favorite the .22-250.