-always require deposits before giving them the keys and do full screening(background and credit) before hand/check their referrals.

I took a check one time, because I happened to be at the property and was going to the very bank that the tenant was on their way to for the deposit. 75% of our tenants pay electronically online, and the remaining 25% go to a local bank where we have a tenant access account set up and make their deposit that way.

Up your liability insurance and require a hefty pet deposit. 

We are dealing with a property now that the tenant failed to meet that standard and we are assessing a fee from the security deposit to cover the expense.

I'm trying to get together an Itemized Security Deposit Deduction Form and then maybe take them to small claims court for unpaid rent.

I paid an initial 19K deposit (1/3 of the 56K total SoW).

As far as how you handle last months rent if she pays until re-rent Month 1, month 2 then you return last months rent because Month 2 is the last month and the security deposit minus deductions. 

Or 24 months bank deposits?   

I get an extra 35/month per dog and an extra 600 on the deposit that is refundable.  

If you're not willing to waive it and potentially forfeit the earnest money deposit.