What must children do regarding their father who accuses their mother of adultery, mistreats her, and threatens them, taking into consideration his age and the distress they face? And is the mother considered rebellious (nashiz) if she abandons the marital bed because of that?

You mentioned reprehensible and dangerous matters about your father, including: accusing your mother of adultery. If he does not have proof, then he is a slanderer, and slander is a major sin. And his saying to his children, "You are not my children"; if he intended thereby to deny their lineage, then this is a grave sin.

Despite all this, your dutifulness towards your father remains steadfast and his wrongdoing does not negate it. Therefore, you must be dutiful towards him and treat him kindly, no matter how badly he behaves. And among the greatest acts of kindness towards him is offering him sincere advice gently, along with supplication.

The same applies to your aunt if she behaves in a similar manner; she should also be advised and reminded to fear Allah the Almighty.

Your mother must obey him in what is good. It is not permissible for her to abandon him in bed, even if he is unjust to her. If your father persists in this state and advice does not benefit him, then it is her right to seek divorce or khul', but she is not obligated to do so. If she does not see a clear benefit in separation, then let her be patient and pray for his rectification.