Is a mother sinful for refusing to sleep with her husband due to his constant abuse and verbal threats to his family? And is his supplication against them accepted? And what should the children do in this situation?

If your father curses, slanders, invokes curses upon his children, and mistreats his wife, then he is of bad character and in violation of Islamic law, as the Prophet (peace be upon him) said: "The believer is not a slanderer, nor a curser, nor an obscenity, nor an abuser," and his saying: "A Muslim is he from whose tongue and hand the Muslims are safe." He also forbade invoking curses upon children, and Allah commanded treating wives with kindness.

However, his bad character does not negate his right to dutifulness. You must treat him with excellence. If he strives to make you associate partners with Allah, then do not obey them both, but accompany them in the world with kindness. Your adherence to the limits of Allah will prevent you from being harmed by his curses or his displeasure.

As for your mother, she must obey him in what is good and advise him. If he does not change, she can refer her matter to someone who can influence him. If the harm persists, she may seek divorce, and enduring that has a great reward. However, she is not permitted to refuse his bed if he calls her, due to the hadith about the angels cursing a woman whose husband spends the night angry with her.