The .17 Hornet might be Your best bet for taking down the many Furbearers Your targeting...as several mentioned Your choosing a pretty broad selection of Animals that might require a little larger Caliber especially when getting up to Foxes & Coyotes.
I have a .22 Hornet and have shot several Groundhogs and it didn't seem to do a lot of pelt damage...but I didn't skin it out so I couldn't say for sure how bad it was?I can say with the Hornady 35 grain ballistic tips it only had a small entrance hole and NO exit hole!
I'm still waiting to call in a Coyote and put it down with my Browning Micro Hunter .22 Hornet...personally I think it will do the job with good shot placement and under 125 yards,if not I rely on my .204 or .223 for larger Predators.