I wish I could have a couple of mouser cats but the dogs hate cats. They don't mind the ducks, chickens and guineas but they will attack a cat. They think of cats as predators and will destroy them. I used to have chickens that loved to go after mice but the ones I have now think of them as their pets. I've seen them eatting out of the food bowl with a mouse sitting in it.
I haven't seen any evidence of our chickens chasing rodents, for sure. We don't have any outdoor dogs. We do plan to get started with Nigerian Dwarf goats for dairying, and then we'll need a Livestock Guardian Dog. But our LGD(s) will have to accept cats, no two ways on that one for us, being cat friendly as we are.