There are clever and expensive feeders called treadle feeders. The chickens learn that when they step on the treadle a lid opens and they can eat. It keeps rodents out of the picture. Like I said they are expensive....saw a wood kit at the tractor supply store for $50! crazy. I've printed off a pattern here on BYC and am going to try my hand at building my own. Cats will coexist with chickens quite well and keep the mouse population down. Also I keep my bags stored away from the coop in metal trash cans in our garage and near by plugged into the wall is an ultra sonic contraption that suppose to annoy mice by putting out a sound only they can hear. I have to say I haven't seen a mouse in the garage all winter and I usually was seeing them all the time the winter before. I haven't put one of these devices in the coop though as I'm thinking the chickens might be able to hear that noise too and it would be stressful for them to hear it all the time. I should research it.... Always things to do with this chickening...things to learn!
Ha!
Also should let you know that there are certain breeds of chickens that are actually good mousers themselves! The buckeye is notoriously known to eat mice. Mine just watch them scurry by and regard them as lowly coop mates.