Just as a breif aside, I believe it is a breach of your human rights to refuse to allow tennants to keep pets, especially cats & dogs.
Obviously there should be provisions made for damage to carpets & cleaning etc, which should be made in the contract at the time you take out the tennancy, but I'm pretty sure that it's illegal to point blank refuse.
I'd have to check everything though on that, and it's not likely that they'll allow you to move in if you start negotiations with "it's a breach of my human rights"
It's always worth asking if you're allowed to keep pets, if you can come to some kind of arrangement to pay for repairs / damage, and always offer to clean the carpets when / if you move out. That'd be where I would start. Most property owners are reasonable people, and if you're offering that to start with, most would probably be OK with it.
As to locations, Temple Newsam as mentioned above, is absolutely beautiful.. there you're not too far from places like crossgates, which is a fairly cheap area to rent, as far as I know anyway
