I will catch or kill every coon I can around here too. Besides the rabies, they are the most destructive of predators with the chickens... they will kill for fun and a family can wipe out a bunch of chickens in one trip. Had them tear up pens of young birds, ripping wings off, legs, strewing the birds around in the field/woods. They also get into the corn here and will strip ears and eat one bite and go to the next. Hate them with a passion. If I catch them in the live traps, there are a couple people that will come and get them... use them to train their coon hounds with. Fine with me. If I hadn't been the recipient of them being so destructive with the chickens, and killing for fun, I probably wouldn't hate them so much. Wanton killing and not eating it just gets my dander up.
At least the fox will carry it off and they do have some problems with mange and some rabies here also. The possum will eat it... although I try to get rid of any of them due to the diseases they can carry. But I don't go out of my way to kill them unless they are being a menace here. I lock up the chickens at night so they have some protection.
The other problem is the hawks and especially the eagles getting the free range chickens...
The coyotes are smart enough to be a worthy opponent... when we find them getting in the sheep, then they are fair game... and we will hunt with night scopes and whatever it takes. But if we don't have any problems, we don't go out of our way to kill them.
We have a very healthy turkey population, and more deer than we can deal with... they actually do more damage in the hayfields and the fall grains/cover crops due to the numbers.