What is the ruling of Islamic law regarding a father severing ties with his children and forbidding the mother from seeing her son if she communicates with him, all because the father refused his son's marriage to a girl with a bad reputation? And is it obligatory to boycott the brother in this situation?

The brother must abandon marriage to a woman with a bad reputation. He must obey and honor his father and search for a woman of good religious commitment, character, and reputation. You all must advise him to do so. If he refuses the advice, the father may shun him and order you to shun him if he deems it beneficial for him to be deterred. You should obey your father if this will deter him. This all applies if the girl truly has a bad reputation and it is not merely rumors. As for the father forbidding your mother to him if she speaks to your brother, the preponderant opinion is that this is ẓihār.