THe Tur ends the siman by saying that you can hunt for chayah game be-tzin'ah, but not in public. How does one hunt for a chayah privately? Aren't chayot by definition wild and not domesticated? Or are we talking about a deer in your enclosed and private backyard? But if so, is that really hunting?

I thought things like killing and trapping mosquitoes go into the halachos of hunting?

Privately owned land.

The house is privately owned.