Our offer was not accepted right away: We received a counteroffer, which we agreed to. The counteroffer involved a little more money, but not too much. The main change was that they want to close escrow in 12 days, so we need to move fast.
A friend of mine (who will now be a block away from us) has a cousin who is a good friend of the family that lived there for forty years. The husband was a contractor. When he died, his wife sold it to her own son, and he rented it out to his sister until recently. I am therefore confident that we will not find any surprises during the inspections.
I am feeling very optimistic that this deal will not fall through. But boy, there is so much to do in so little time. It will be stressful, but stress of the good kind.