It certainly could improve your cash flow but it won't help with finding prospects, screening tenants, maintenance, deposit dispositions, etc. 

It usually starts with a notice being posted, so get that done asap to 'start the clock'.Keep in mind you should receive their security deposit as a transfer from the previous landlord as a part of closing, so that will at least compensate you somewhat for the lost income if they haven't trashed the place and such.One other set of questions would be - do you want them to leave?  

(Had a tenant put a deposit down and has taken a month for the PM to put in a fridge and stove. that was paid for a month prior to that).

Does this mean that we will just pay less at closing and not get the physical security deposits?

Rent is rent, deposit is deposit, and I suggest keeping it that way.

How have you handled the transfer of all deposits/payments received and the remaining payments received after closing on those future bookings?  

If so, make sure agreement is in writing, signed by both parties, specifies new end date and ensures any existing security deposit will still be dealt with properly per landlord/tenant laws, and I'd also ask for a partial payment to be issued at signing to make obtaining his new place easier.  

Just download all your transactions monthly to excel and just code them there (rental income, security deposit, insurance, taxes, and etc).

If you get a deed on a rented property are you then on the hook for security deposit, etc?

I've read that I should open a business savings and checking at a local institution - particularly for the security deposits.