I used to have barn cats and I have mixed feelings/thoughts.
On one hand, yes, they will help reduce the rodent presence in and around the coop/run. Heck, there's a neighbor cat who swings by my yard every so often, and I know he's been around because I will see neither hide nor hair of any rats for days. (The rats are currently hanging out because I have chicks and food is finding its way to the ground. In a few weeks we'll be back to just treadle feeders and they'll be out of luck!). One of my cats really liked hanging out with the chickens and they were fine with his presence.
But something to also consider is that cats do a tremendous amount of damage to local wildlife like birds and rodents like voles, and with my barn cats (and one in particular) that was true no matter how much I fed him. If you can manage the rodents by using exclusion methods (hardware cloth, treadle feeders), that will be overall better for all the critters in your area.
That said, adopt-a-feral programs help alleviate some of the issues with strays, so if you do go with getting a cat, this would be a good place to start.
And I'm not sure about locking it up with them. At least not all the time. There were times I thought about putting one of my barn cats, Oliver, in with them at night (which is when the chickens are settled and the rodents come out to play). Oliver always liked the chickens and would go hang out in the run when it was open. (He also got very good at eating their food!). Despite him being about 12 pounds and the chickens being 3 pounds, he was always much more scared of them than they were of him.
On one hand, yes, they will help reduce the rodent presence in and around the coop/run. Heck, there's a neighbor cat who swings by my yard every so often, and I know he's been around because I will see neither hide nor hair of any rats for days. (The rats are currently hanging out because I have chicks and food is finding its way to the ground. In a few weeks we'll be back to just treadle feeders and they'll be out of luck!). One of my cats really liked hanging out with the chickens and they were fine with his presence.
But something to also consider is that cats do a tremendous amount of damage to local wildlife like birds and rodents like voles, and with my barn cats (and one in particular) that was true no matter how much I fed him. If you can manage the rodents by using exclusion methods (hardware cloth, treadle feeders), that will be overall better for all the critters in your area.
That said, adopt-a-feral programs help alleviate some of the issues with strays, so if you do go with getting a cat, this would be a good place to start.
And I'm not sure about locking it up with them. At least not all the time. There were times I thought about putting one of my barn cats, Oliver, in with them at night (which is when the chickens are settled and the rodents come out to play). Oliver always liked the chickens and would go hang out in the run when it was open. (He also got very good at eating their food!). Despite him being about 12 pounds and the chickens being 3 pounds, he was always much more scared of them than they were of him.