I've probably trapped 150 coons and 50 possums in my chicken house over the years. The possums and coons both like eggs, but a big coon will go for the birds themselves. Two years ago I had one that came in during a period of blowing snow. The snow would always blow over the coon tracks so I didn't notice until I counted my birds and realized I was 4 short. I trapped with the largest Havahart trap, which had always worked before, but this guy was a monster.. tore the trap up. Then I resorted to an older method which was a wire dog kennel, nailed to the coop opening, with a drop trap door mounted on the outside. Put a trip plate on the inside and a peg through the drop door outside. I put a chicken in a small cage on the far side of the dog cage, visible to the coon, to attract him. I caught the coon, but the darn thing was so strong, it pulled the cage loose from the nails, and managed to slip outside with another bird. Lost three more birds. When the snow melted enough, I found the carcasses of the birds in various places around the property. I've never seen him again. My wife saw him one night, thought he was a small bear. My guess is he didn't get as big as he was by being stupid. But believe me, from then on I went around my coop, which is large, and tightened everything up. Then I put an automatic door on the coop, made of salvage garage door parts. I can open/close the door with a controller (real handy when the weather's nasty). The door is very secure now, and we take great care to see that it is closed right at dawn.
Re: dispatching of critters. Typically I shoot them. You can buy lo-noise 22's and they'll do nicely for small critters (except for possums -- takes a shotgun). If you shoot indoors, with the doors shut, during a noisy period of day, no one will probably notice. I hate drowning critters, but it's about the only safe thing with skunks. Skunks can't spray if they can't get their tail up, so if you can get them in a small enough trap, drowning is the best bet. If they are bigger than that, be sure to have a long rope attached to the cage, stay upwind, and drag them to a spot where you can use a rifle. If you are going to keep birds, be ready for anything. I had a mink eating my birds last winter. They are picky eaters, go after the breast meat at the base of the neck. I've also had bouts with weasels. They like to just bite the throat and drink blood. To catch them you need something bloody inside a box with the right size hole, and a rat trap screwed to the bottom of the trap. Put sprinkles of the blood around the box, but put the main attraction inside, on the other side of the rat trap. Build the box right (you can find designs on the net) and if successful you'll see their butts hanging out the hole when you check in the am. The coyotes around here are plentiful, but I think my Great Pyranees peeing on his territory has probably kept them at bay.
You can't be a softy about this. I've invested hundreds in feed and brooding and hatching only to have hens just ready to lay, and then have gone to the chicken house to find several dead birds because of predators. I have one neighbor (a newbie who lives just across the creek from me) and she thinks coons are "darling" and I'm a monster. She raises them. The closer the general population moves to me, the worse it gets. I've even invested in a high power airgun that shoots as well as a .22, and has a scope, and is pretty silent, so that I can do in critters that are preying on sweet corn and other goodies I've worked for years to grow. I've knocked coons out of trees with it. Ordinances are different about air guns. They aren't classed as "firearms." I suspect it won't be too long, and some group will pass an ordinance against my activities in our county.
There's even a hound dog that was running off with some of my expensive prize hens up the road. His owner said "oh, well, that's just how dogs are..." I told him (and we do have leash laws in our county -- which I observe) that if I saw his dog around my coop again, HE'd never see it again. You wouldn't believe the runaround I got from the sheriff, when I asked him about the laws. The sheriff's secretary said with a wink was "well, if no one knows what happened to the dog.. it's easier all around." Problem is, I feel bad about the dog... it's the owner who's really doing the wrong thing.