What is the ruling on dutifulness to a father who is extremely angry, who shouts, swears oaths, breaks things, threatens harm and insults his children with obscene words, and is it permissible to keep away from him?

There is no blame upon you for not angering or disturbing your father, and it is obligatory upon you to treat him with kindness and righteousness, even if he is a polytheist. If your moving away from him pleases him and is better for you, then there is no blame; otherwise, strive to be patient with what you encounter from him. Your patience with your father's harm and insults, seeking Allah's pleasure, is more worthy than an employee's patience with his manager. And if your father dies after your patience, you will not regret it. You can inform your father that whoever strives against himself and does not become angry will have Paradise, and that destroying furniture is a waste of money and forbidden, and that the Prophet, peace and blessings be upon him, was pure of tongue, so he should follow his example. Do not advise him when he is angry, but rather after he calms down, and let whoever he is angry with quietly leave his presence. Defend your mother calmly without fighting your father, by reminding him of Allah and seeking refuge from Satan.