First, I am assuming this is an approved short sale.
The lenders do provide a formal extensin letter when close of escrow date is extended. So, yes, do ask the listing agent to give you copy of the exztension, otherwise, proceed as it has to close by 02/15 - if there is no extension and you can't get the loan funded priorr to that, the house might foreclose; or the contract has to go back to be renegotiated (BTW, some of the settlement costs will be different for a new closing date).
If the seller wants to stay after you close escrow, there should be a formal agreement drawn up - for less than 30 days rent back use CAR Purchase Agreement Addendum (PAA), there is a paragraph talkes about rent back less than 30 days. If it's a longer than 30 days rent back, you will have to sign a 'Residentail Lease After Sale' Addendum.
Check wit your Realtor on the best way to approach.
Frank Howard Allen Realtors