You should absolutely bring your pet! I think cats, generally speaking, are easier than dogs (I have one of the latter who's been such a source of comfort during times of stress). You can leave them home for longer periods of time, but you may also be spending a lot of time working from home. As for funds, it just comes down to budgeting. I figure I spend up to $80 a month on average, including vet bills (but this amount is split between two caretakers). This isn't bad at all... it means I eat out a few less times a month, which is fine. I stay home and cook and spend time (and share table scraps) with my pup instead!
I empathize with the fact that pet-friendly housing is more limited. However, I think renters often view grad students differently from undergrads. One thing you can do is prepare a 'pet resume' (sounds corny, I know, but it can be helpful). Just prepare a brief page with stats on your pet: size, temperament, level of activity, how long you've been with the pet and how you intend to take care of her. Include references for the pet -- previous landlords, vets, anyone who can vouch on behalf of its behavior. Offer to put down a pet deposit, if necessary (up to half rent? Where I'm at, most places wanted about $200-$500 pet deposit for $1000-$1400 rentals). I don't know the legalities of non-refundable pet deposits in other states (sounds a little iffy to me) but it's a good faith offer that could sway potential landlords.