We have had cats(down to one) and dogs(down to one as well), we have never let our cats outside.
We do have a "feral" neighborhood cat who had a litter, so I trapped them all, including momma. Took the kittens to a buddies farm and had the momma spayed, released her back to the neighborhood and have no real problems.
We have two neighbors who let their cats(1 each) roam, but the one I had spayed keeps them away, this is her territory. She lives in a run down garage my neighbor has basically abandoned.
Additionally, I have never encountered a mouse on our property in almost 6 years. That said, we have a significant amount of birds in our yard, never found a dead one. Maybe my bird feeders are too high?
Now, if you want to talk about squirrels, those things are a real nuisance.