Is it permissible for my mother to accept a marriage proposal from my husband's father, who suffers from muscular dystrophy, and to help him in exchange for a dowry and property, and what is the ruling on his disposal of his property during his lifetime to guarantee the rights of the heirs?

There is no objection to your mother marrying your husband's father. His payment of the dowry is valid and does not diminish the rights of the heirs, for he has the right to dispose of his own money. As for his gift to his youngest son and his spending on him, if it is due to the son's need, there is no objection. But if it is a preference without justification, then it is not permissible except with the consent of the other siblings.