I ask every renter if they are traveling w children or pets. Not suitable for children at any of our properties as someone tried to sue us after they dropped an infant at our house. We were obviously not found at fault as this is just ambulance chasing at its finest, but it was a waste of a lot of our insurance company & attorney's time. I ask the size of the pets. We have rented to dogs in the past but we could never do cats there's no reason for that...
If you are nervous about the rental, ask if they would agree to sanitation costs at $600-$1000 furniture/shampoos - most people find another rental. There are tons of pet friendly rentals on here!
I had someone ask me to bring two 60 lbs Italian Mastiffs...you know they drool & tear everything apart! Would be impossible to chase someone down for damages like this.