What is the ruling on this sale, and is it considered void given that the agreed-upon amount was not fully paid within the specified period, especially since one of the brothers says that the sale was concluded with the first agreement (800,000 Syrian Pounds)?

If your mother canceled the first sale with her consent, then the second contract is valid and binding. However, if she did not cancel it with her consent, then the first contract is the valid and binding one. The condition that you must receive the price within two months, otherwise the sale is void, is a "cash option" (khiyar al-naqd), which is a valid condition that grants you the right to annul the sale. Jurists have permitted it due to people's need for it. We advise you to be forgiving and lenient and to satisfy your mother, due to the greatness of her right.