My quick perusal of the Idaho hunting regulations suggests that you could course coyotes in Idaho. The only big game animals where dog hunting is allowed are bear and cougar. So wolves would be out. However, coyotes are explicitly exempted from the list of big game animals, so absent finding any specific prohibition, I would assume that dog hunting would be allowed on coyotes. If I were going to course coyotes in Idaho I would check with the Fish and Game Department to make sure my interpretation is correct.
Was referring to Wisconsin and deer. Though you can use them to track a wounded animal. You could be right about coyotes, they’re a year round season here. As are boar, though there aren’t that many. Doubt they have regulations about pests. But you’re right, one should check state by state. Don’t know anyone who hunts raccoon, usually trap them, but dogs might be ok for them.