Of course, I tried but she didn't give me any rent before the closing, and I prorated the rent and sec. deposit from day 5-30. 

Make it clear that if the house is left in good shape they will get their deposit back (but if it's in bad shape or rent is due that will be deducted). 

I"m keeping their deposit

I offered him 1 month free rent until we fix the issue, he refused and says he will return keys when he gets his deposit back.

Even if you somehow got $80k deposited into your checking account before closing, a conventional lender is going to ask to see where it came from. 

I have a 4k earnest deposit, then the costs of the inspections and appraisals, as well as about $800 remaining for the downpayment, so I don't have too much into the deal cash wise, but I would rather not lose my deposit.

If you don't mind the cat, but need another pet deposit, then ask for that. 

IF someone doesn't even have the money for an earnest money deposit then they are not likely to be able to pull off the deal.

Beginning with rent for the month of September, please pay your rent by depositing the entire amount into the following bank account at (your bank): account #123456; routing # 123456.

Then I require a double deposit, plus an additional fee to cover a mandatory, professional cleaning after departure.