You are all full of questions.
As a youth, I had use of Dad's 22 LR singleshot. I guess I shot everything that was legal to shoot with that rifle, from mice to porcupines.
When I was in highschool, I bought a Ruger Single Six and really learned to shoot with it. I built a holster and took it everywhere except to school. I guess I shot more critters with that six gun than any other gun I ever owned. Flies, grasshoppers, mice, rats, gophers, muskrats, squirrels, cottontails, jack rabbits, porcupines, skunks, and bobcats, to name some. I have never taken a shot at a fox or coyote. I quit using Dad's single shot, after getting the Ruger, and never bought another 22 until many years later. I got a Marlin semi auto when my daughters and wife decided to take hunter's safety ed. They needed a rifle to shoot for the test. I have never hunted with it.
Ammo? I just bought the standard velocity stuff, Nothing fancy. Maybe hollow points. It didn't seem to matter much, if you made a good hit the game was yours.
I used to hunt bobcats with dogs and the sixgun was the gun I always took. One shot through the lungs always brought down the treed cat. It would not jump out of the tree with a lung shot, but after a minute or two it would bleed out and fall down dead. This was good as it would not claw and chew up the dogs.
Then comes the obnoxious cat that won't go up a tree, but wants to stand off the dogs on the ground. Usually an old male. Well, the dogs will eventually wear it down till it rolls over on its back then it becomes dangerous to the dogs as one will grab it and it will grab the dog usually by the head or neck, and start raking the dog with its claws of the hind feet. I didn't want to shoot for fear of hitting a dog, so I waded into the fracas and hit the cat a good stout blow over the noggin with the barrel of the six gun. End of cat and dog fight. On two occasions I killed a bobcat in this way with the Ruger Single Six.