The rent and deposit will already cost that much depending where you located.

Offer and deposit submitted and accepted by 8AM that morning. 

Before closing, I made sure prorated rent and security deposits were included on the HUD; however, the seller shows up at closing and said they just found out that tenant B had not paid their rent this month per the property manager. 

Regardless of your plans, get the tenants to sign estoppel letters verifying their security deposit, rent amount, etc.

Normally we write an "offer to purchase" with an earnest money deposit, which when accepted, opens up a due diligence window of 10 days.

Basically, after signing the contract, I would either dive in and hope for the best or lose my $5000 deposit.

You can charge them a rent and sign a lease for 30 days max taking a deposit.

If it's closer to $1200 than $1100, it might be worthwhile to see if you could offer them $500 + their security deposit back (which you'll probably have to return anyway if the previous owner didn't keep good records of the move in condition, which most don't) and see if they'd be willing to move out. 

If they say yes , you sign a contract , pay the deposit and get on the list .  

ADP, Paychex, Gusto, etc) that are priced reasonably, will keep you compliant, and will have that money deposited into your checking account automatically with a W-2 issued in January.Not sure what you mean by 'handling your distribution'?