Does anyone know if it's possible to have a cat in one of the shared apartments, or do I have to get a single?
Unspoken/unwritten rule is that if your roommates don't complain and you're in an unfurnished place, no one cares about pets (provided you're not in Lenfest/Lionsgate, which are newer and nicer and they tend to be more strict about). I just emailed my roommates as soon as I got their info to make sure they were okay with it. Allergies aside, it seems like most people relish the opportunity to have a pet without any of the responsibilities.
Also I pay $1000 for a shared room in a large 3 bedroom, which is common, so $1300 is definitely on the high side for UAH (maybe that's closer to what you'd pay in law school housing), and sharing an apartment has worked out fine for me so far. I understand if you're older and have lived on your own for a while, but sharing an apartment is not that big of a deal. These aren't dorms, they are regular, average-sized apartments that are similar or better than how most people in Manhattan live.