What is the Islamic legal ruling on dealing with a mother who abandoned her son in his childhood and treated him and his family poorly, then started forming suspicious common-law marriages at the age of seventy, while desiring to live alone and refusing to live with her son?

You must fear Allah and strive to be dutiful and respectful to your mother and treat her with kindness. For Allah, the Exalted, has enjoined honoring one's parents, even if they are non-Muslims. And the mother's right is more emphasized than others. Part of honoring her and treating her with kindness is to advise her, guide her, and direct her to what benefits her in her religious and worldly affairs. You must strive to prevent her from deviation, and it is not permissible for you to abandon her, as cutting off ties of kinship is among the greatest sins. As for the unregistered marriage (zawaj urfi), if it fulfills the conditions of marriage and there is no legal impediment, then there is no religious objection to it.