If you know anyone with Jack Russell terriers, they would be glad to deal with your rodent problem. If you search on BYC, I'm sure you can find the
"Water bucket mouse trap.", or get some of those electrified mouse traps that zap them. You could put out after the birds are safely cooped for the night.
I have a rat problem as well. I tried snap traps when my chicks were in their brooder so I didn't have to worry about the chicks getting in them, the traps were also in the run so none of my other animals could get hurt but I did trap a squirrel and a wild bird accidentally
. That dwindled the population some, I know you won't be able to use that option though but you could put them maybe up off the ground like up high in the barn where other animals couldn't get to it?
I have an electric trap that works really well for me. I get a lot of rats in this trap and it's easy disposal, no touching rats whatsoever. I wiped out all the dumb rats but the smarter ones caught on so I just have a couple left. Rats are hard to get rid of but the electric trap is really something you should try.
I've tried poison up in the roof and the rats are it, the container it was in, and are still around... So much for that.
Another thing to try is a weasel box with a snap trap in it. Look up weasel trap and it'll show you how to make them. Just load them with food and it's convenient because none of your animals can get in there.
Sticky traps could work but they're not kind to the mice/rats and it wouldn't kill them, you'd have to do that.
Hope I helped, mice and rats are hard to get rid of and they get smarter the more you catch. Hopefully one of these techniques will work for you!
How old/big are the chickens in question? The cat will probably try to go after your smaller chickens a few times until it learns that it doesn't get to eat them. Our cat originally stalked our chicks till they were about 10 weeks and started chasing her, pecking her on the nose, and generally keeping her in check. Now they all get along and have a mutual understanding that neither are dinner.