Rats can spawn in claims, they eat crops and are disturbing cause they stop you when you (or your bot) is moving to them.
But you can get a cat.
He kills rats on sight, a cat is bond to a claim, not a person, he have no "hitbox", like a flying bird.
You can reskin your cat(s) with skins you buy at the cash shop, if you want more then 1 cat/claim you can buy a "place'" at the cash shop.
How to get them ?
Duno, perhaps someone can tame a lynx ? Its hard, its a statut vanity thing you trade, or it can be a quest line so anybody can have one if they put some time on it ?
When claim vanish? you got a wandering cat you can capture like a wild horse but with a fish, then right-click on your claim totem to put him on your claim?