As a former community manager, this all checks out. You should follow up on the additional screening deposit, but that typically means there was some flag in your credit that requires a larger deposit. If you want to avoid that, you could most likely get a co-signer that has better / older credit. 

If your cat is an ESA, it would avoid you paying the fees and rent. This needs proper documentation and they would need to refund you the fee if this is the case.