In my experience buying properties using conventional personal financing, my lender required two months of bank statements with an explanation of any large deposits not coming from a W2 job.

If your bank has a location near here might be worth giving them prefilled deposit slips with their address on it so you know who deposits what. 

Since then, they've put down a deposit to hold (got that from B Turner's book on managing rentals). 

Who's security deposit does the wall repair come out of? 

We have someone we'd like to rent to and are working with them re: there first, last and deposit payment.

First, any security deposit he has, could likely be shifted to down payment at closing. 

You may get lucky and find a renter to pay that but most renters will just move out, forfeit the deposit, and leave you trying to chase after them to collect.

Then you can decide what that practically looks like as far as provide deposit refund, maybe being flexible on move out date etc.

Now, if you were to find something BIG during that inspection that was going to be a nightmare if you purchased the property, what you could do, is simply walk away from the deal and the Seller would just keep your earnest money deposit.

Not due a deposit.