Do your cats have access to the chicken coop at night? It sounded like it was closed off.
I think part of why you have so many is that they avoid or are chased off of your neighbor's property.
When we bought our previous house on property, the metal building was full of rats/mice. You'd go in there and they would stop and look at you and continue on with their business. A handful of cats later and we never had a rat/mouse problem again.
As a somewhat related aside, we stopped all burrowing animals from entering our coop by digging a foot deep trench that went a foot away from the coop. I then took a heavy wire mesh stapled it to the base of the coop, rolled it down a foot, then away from the coop a foot. So we had a 1'x1' L at the base of the coop. I then back filled the trench. I would occasionally find a hole dug next to the coop, but it always stopped at the horizontal wire mesh a foot down.
A perfect gun for this type of work is the Gamo .17 or .22 caliber air rifles.