Which pets are excluded often comes down to the prior bad experiences those landlords (& their insurance companies) have faced in the past- that determines how open-minded they'll be.
As far as cats, when I was in college, I once lived in an apartment where someone was feeding a stray cat that wasn't allowed as a pet. As a result, the entire building got infested with fleas in the carpet, and had to be fumigated- talk about miserable, & probably costly. If everyone was responsible with the animals they keep...but sadly, they just aren't, so rules get made for all. Hang in there.