She doesn’t sound psycho to me either. She’s accommodating you by allowing you to bring your cat – whether it’s because she likes you or whether she recognizes that you have an attachment to the cat – either way, she’s being pretty nice about it.
She might have had bad experiences with pets before, or maybe she’s heard horror stories.
I don’t see anything wrong with them using part of your deposit to clean after you leave. It was nice of her to tell you in advance.
I think you’re borrowing trouble, extrapolating that she’ll be difficult when she’s been anything but.
As a current tenant AND landlord (yeah, of different places ;)) I agree with most of the posters that this lady sounds pretty reasonable - except in one thing I’m going to have to disagree with the prevailing sentiment. I don’t think it’s reasonable to say you’ll take money out of the security deposit for cleaning “because of the cat”.
I don’t know what the norms are where you live, but my expectation is when you leave a place, you clean it. IF you don’t manage to do this to an acceptable standard, THEN the landlord gets to take your security deposit money to pay someone else to do it. Doesn’t matter if you’ve got a pet or not. It may be harder to clean if you’ve got a cat, but that’s your lookout, not hers.
When we left our last place, we steam-cleaned our carpet to within an inch of its life - it was the only one in the block that didn’t need the professionals to go over it later (everyone in the bottom two floors got our marching orders at the same time). I would have been most peeved after all that effort if they’d taken money out of a deposit for cleaning “because we had a cat”, because that sucker was clean enough to eat off.
Of course, if I knew that my landlord was planning to take my deposit anyway, I just wouldn’t bother cleaning it properly in the first place…
The rabbit that I’ve had for the past three years was put up for adoption this past weekend through a rabbit adoption nonprofit, solely because of this move.
Since I feel so strongly that pets are not disposable and that once you take ownership of a pet, you have it for good until the pet dies, I feel like a complete hypocrite and this has been incredibly difficult for me, moreso than I can convey in a post. However, my reasons for moving are far more imporant than my reasons for staying where I am (safe school for Hallboy, no more gunshots in the neighborhood, no one in the alley behind my house smoking the crack they bought around the corner, no convicted sex offenders living within steps from our front door, etc.).
That was a joke, son.
(And yes, I realize you are a woman. 
)