I'd also advise putting the burden of dealing with security deposit on the tenants.

Check your statements to see who has the security deposits.

Would you be able to pay the security deposit of ($ amount) at the lease signing?

Anyway, he stated that if I wanted to lock up the deal, I would have to put down a $1000 deposit, Purchase & Sale with an additional $5000 and all the money within days. 

There are lots of ways to mix and match and get what you want.

Anyway, we decided on a refundable deposit, proof of it being a "service" animal, and an addendum that she sign and return that basically protects both parties.

When you get the checks take a photo or copy of each one to document who it came from and deposit them into your account. 

a specific quote from the actual HUD guidance,"HUD is providing this guidance to help housing providers distinguish between a person with a nonobvious disability who has a legitimate need for an assistance animal and a person without adisability who simply wants to have a pet or avoid the costs and limitations imposed by housingproviders’ pet policies, such as pet fees or deposits."  

I'll ask for a larger deposit to mitigate some of the risk.

Pull up 10 or so of their "Delinquent" records.Go to the county tax website and look up these properties to see If the Delinquent data from your provider matches the county Delinquent data.Be sure to report back here with your findings!