We get mice indoors occasionally and to be honest cheese never works in a mouse trap, as you say chocolate or another thing they like is peanut butter. Tom and Jerry keep bringing mice home to play with in the garden then my dog Ruby gets involved a little Border Terrier, Quite funny watching...
Had cats all my life and some have never brought anything home, some have brought the occasional thing home. The 2 cats I have now ( Tom & Jerry) are at it all the time, I've seen them struggle over the fence with birds nearly as big as them! The one thing they won't go for though is a seagull...
Wild dogs yes, They don't use wild dogs on fox hunts otherwise they would be too busy eating things like Rabbits to bother chasing after Foxes. That has been trained in to them.