To extend that income you can put a lease and security deposit in place with a new tenant at home #1 and some lenders will count that towards your income to purchase home #2.

I had a $13,000 check deposited four weeks prior from a client of mine.

Also make sure you have them pay a deposit and sign a lease right away because they will usually keep looking.  

If you deposit your cash this might work.

I usually just keep the security deposit which they seem to expect.

I keep two banks, I keep my deposits, insurance premiums, taxes, mortgages, and CAPEX in one bank. 

Other than 1) Actually firing the PM, 2) Having a separate bank account to hold the tenant's deposit, 3) Having the old lease end and new lease start the next day...Are there any other considerations I need to take into account?

At first I told them that a 14 day deposit hold is $100 non refundable and put towards there security deposit, and we could make the deposit to hold extended to 30 days for a additional $100 non refundable fee that does NOT go towards anything (accept my pocket).

It will be April sooner than you think.Often it comes down to being able to pay the security deposit on the new place before having gotten it back from the old place.

If the down payment is being applied to the purchasing price, i am not sure why people would be concerned about it.. all it really is a deposit you put for an apartment security deposit as well, but it could be because its a big amount of money given forefront and because if anytime during the due diligence period, something falls through and “potentially” not being able to get that money back...?