I truly believe hunting is the worst way to control predators. By and large, your resident predators usually respect you and won't take your chickens. They will also keep away other predators and somewhat control the rogue ones. If you hunt and kill the resident predators, you invite the rogue...
Well, I wasn't talking about a coyote, which I would not catch to start with. IF a coyote came on my property, I'd shoot, not catch (however, they are around but stay away; coyotes are a big problem here too just not at my place) -- deterrence works better than eradication. Dogs & donkey work...
Depends upon where you release. I live in a National Forest and we have plenty of locations where there are not people or not people with chickens/ livestock. In fact, most people have some horses and call it a farm. There is room on the planet for the billions of us and a few animals too. Wild...