Sometimes the mere presence of a cat will keep mice out of the house. The cat doesn't have to be a hunter; the scent of a cat scares the mice away. But I suppose it depends on how cold and hungry the mice are in any given winter.
My sister-in-law once saw her dog and her cat posed in the kitchen door, watching something with great interest. She checked, and they were watching a mouse with its hind feet clamped on the edge of the cat's dish, drinking its milk. The pets evidently thought they were watching "Wild Kingdom."
If you want a nice pet, get a kitten from a mother that was a nice pet. Make sure the kitten has been handled and is used to people. Don't adopt a feral animal. Those are tips that may not guarantee success, but they will stack the odds in your favor.