seller rent backs are very common in CA so common that there is a section in CAR Form Purchase Agreement Addendum ( PAA) just to address this issue at any rate, you need to be contractually protected
i would not allow the seller to stay in the house after closing. Too much liability issue.. if the falls during his move, he may end up suing you because he became a renter and you own the property.. if he decides to over stay in the house, there is nothing you can do to get him out. It would take months of court proceedings... i would politely refuse his request by saying that after the closing, you don;t have a place to live yourself. Delaying closing date would be a better option but never let the seller stay after closing.