Is a patient with schizophrenia allowed not to speak to her mother and to avoid discussion with her, and will her mother's supplication against her be answered despite her mother's ill-treatment of her and her preference for the brother who mistreats her?

Patience with a mother's harm is obligatory, and it is not permissible to abandon her. Rather, one must address her gently and avoid provoking her anger, for disobedience to parents is one of the gravest sins. Your dislike of her actions will not be held against you by Allah, and her prayer against you unjustly will not be answered. As for your brother cursing and hitting his mother, that is an act of disobedience (to parents), and he must be advised and shown the gravity of his actions.