Is the sacrifice offered by the father, with money paid to him by the son, sufficient for all members of the family, or is it obligatory for the son to sacrifice only for himself, his wife, and his son, and does the mother fall under the second sacrifice?

A single sacrificial animal (udhiyah (definition)) is not sufficient for both a father and his son who lives independently. However, it is permissible for the son to offer one sheep as a sacrifice on behalf of himself, his wife, his mother, and his son. As for a brother who lives with his father, the father may include him in his sacrifice along with the rest of the family, as long as they reside in the same house.