Is it permissible for me to live with my husband, and will Allah punish him through our children, given that he has cut ties with his mother because of his father, and refuses to discuss this matter?

If the husband is estranged from his mother, this is an act of disobedience (ʿuquq) and a major sin. He is not permitted to do so due to the mother's ill-treatment or the father's command, because there is no obedience in disobedience to Allah.

He should be advised gently and reminded of the gravity of his action, and prayers should be offered for his repentance. If he does not respond, seek help from someone who influences him. There is no blame upon you for living with him. Sins may be a cause for worldly punishments, but it is not necessary for these to materialize in the children.

Focus on raising your children and guiding them to good.