RE: treatment of wounded animal
You certainly can carry a .22 as a side-arm while hunting for deer in Michigan. You just can't use it for hunting deer. Dispatching a deer is a different thing. I am not sure if it would be illegal to use the side-arm to dispatch the deer but it certainly would not be WRONG or morally suspect. And I seriously doubt that any conservation officer would ticket a hunter for using a .22 while dispatching a deer already mortally wounded by a legal cartridge.