What I have found on google is carpet wear is 3-4 years and then can't be charged for. Ask for a break down from the rental place. Anything not directly in your room could be a grey area. Damages in their room is not your issue pending you can prove which room is yours. Gather up all info you had on the place just in case so it is in one place, but other then that I woudn't sweat too much. Did you leave a deposit and did they have a break down of all bills? Use google for most of this instead of us wanna be EP lawyers though.
Since you're not legally a party to the lease/rental contract, you don't owe anything to them!
I wouldnt pay.They knew you had a cat,hadnt let the place in pristine condition when you first rented,and were already told there was no problem.On top of which they cant be bothered to give you a breakdown.On their bikes!
Take it to a civil lawyer..
Press the issue. Hopefully you took pictures prior to moving in.
Pay up b i. T. C h