80 year old mother – Homevestors

It is legal to market the house before closing on it. It is even legal to enter into a contract of sale before you own it--the contract would not bind your mother, since she's not a party to it, and if the sale of her home didn't go through, she'd have no liability, but her buyer would, because they could deliver the house as contracted. Fundamentally, people can "sell"--or rather, contract to sell in the future--things they do not yet own. (If you think about it, that's exactly what stock "futures" are, for example--an agreement to sell stock not yet purchased; it's also similar to how a car dealer can agree to sell you a car which they still have to acquire from the distributor or manufacturer.)
They could even be paid for the home in advance, if their buyer is willing to pay for a home they can't get yet, and which possibly they will not get (e.g. if the sale from your mother to her buyer fell through). The one thing they can't do is to give possession of or a legal right to the home to another person prior to closing. You can only give someone else what you yourself have, and your mother's buyer did not have possession until the closing of their purchase.