What should I do about my mother who betrayed my father, who is ill with cancer, and seized his money and his house, and insists on her relationship with another man, and forces me to accept the situation or leave the house?

Your duty is to be dutiful and good to your parents, and to enjoin your mother to do good and forbid her from evil gently, without violence or abuse. If you notice any suspicious behavior from her, you must prevent her from engaging in corrupting influences by threatening to inform those who can stop her. If she does not desist, then inform them, while continuously being dutiful to her and avoiding harming her. As for your father writing the house in your mother's name, if the intention is to transfer ownership after death, then it is a bequest to an heir, which is impermissible. If the intention is to transfer ownership during his lifetime, then it is originally permissible. However, there is a jurisprudential disagreement regarding the validity of a man gifting his dwelling house to his wife. Some scholars do not validate it, while others stipulate that the house must be free of belongings not belonging to the donee.