General cleaning or damage deposits can be charged, if all residents are similarly charged.Please remember that a housing provider can hold a resident with an assistance animal liable for any damage the animal causes.Can a housing provider limit the number of assistance animals?

Obviously the previous tenant or home owner are in a fragile state and using force can actually cost you more money in the long run if they decided to damage the house.Now, the story with the buyer.

It's some 14 year old kid in the Ukraine who would do the most damage to our identity.

You might also consider contacting a company like ALE Solutions who help people with temporary housing when they've been displaced due to a covered property damage insurance claim.

Better terms to use might be "fee" or "charge" as they are already understood to be non-refundable.As for what to charge, it might be a good idea to analyse the property and decide what would most likely be damaged by the pet and make the deposit dependent on the repair/replacement cost for that. 

Violation incurs not only penalties under the Property Code (return of all payments made by the buyer including monthly payments) but potential liability under the dreaded Deceptive Trade Practices – Consumer Protection Act ("DTPA") which can involve treble damages plus attorney’s fees.

Any ideas on financing...I'm thinking of possibly seller financing given the owners have not repaired the damages they may not have a ton of money.  

The GRM was under 100, but the tenants turned over far too rapidly (higher average vacancy rate) and the property was older (higher maintenance cost), and the tenants damaged the property as well (yet higher maintenance cost, note broken window syndrome).  

Recently, there was some extensive water damage that came from a busted dishwasher.

It covered up to 6 months of vacancy, eviction charges, and damages.