is it legal (in new york state) for a landlord to charge an extra $20 monthly for a cat? I've already given first and last month's rent as security. If any damage is done, can't they take that amount out of my security deposit?
If the charge is in the lease, probably.
It depends on what NY law allows search for New York Landlord tenant law as Google.
In my state YES. You can charge pet rent and additional pet deposit which can or can not be refunded.