You have excellent posts so far.
The truth is that in real estate only things in writing are true.
When you say "short sale contract was signed" we do not know what that means.
In California there is the Residential Purchase Agreement (RPA), and another related form is the "short sale addendum".
In the RPA in section 3, b) at the bottom of the page states when possession is to take place.
Normally if owners want to rent the property after the sale concludes, they sign a separate lease agreement. The lease payment is normally equal to the buyer's principle, interest, taxes, and insurance payments (called PITI). If the buyers want to occupy BEFORE the escrow closes, there is a form for that.
It sounds to me that the best person to answer your questions would be your Realtor. If they don't know how to handle the situation, then talk with their broker, who supervises the Realtor.
It is common for sellers to rent back, the problem is that the amount of the rent they would need to pay might be more than they can afford.