Is it obligatory for the questioner to remain in his father's house and pay the overdue rent, or should he move to independent accommodation to avoid the harm that has afflicted his family, despite his fear of his parents' anger?

Obeying one's father is among the most important matters in religion. Allah has linked His right to be thanked and shown kindness with the right of the father, and He has commanded us to show kindness to him. Even if he is not a Muslim, this does not preclude obeying him in matters that do not involve disobedience to Allah. What your father has done is incorrect. You must comply with his wishes if it is within your means and the harm is tolerable. Seek your reward from Allah for what you do. If moving to another residence would incur rental expenses, then it is more appropriate for that to be for your father, thereby fulfilling your duty of piety towards him. We advise you to be gentle with your father and explain to him your inability to pay this amount. Try to increase your kindness to him by offering him some gifts, spending more time with him, and showing affection. If you have violated the condition he set for you, then apologize to him.