Is what the sister did, by reproaching her brother and his female cousin for being alone together while she was adorned, correct?

A man being alone with a woman who is not his mahram is dangerous and opens the door to corruption. Your sister was right to rebuke them, while observing the guidelines for enjoining good and forbidding evil, and continuing to advise with a kind approach. As for your brother shunning his mother, this is a great evil and an act of disobedience (to parents). It is never permissible to shun one's parents. Similarly, shunning his sister is not permissible unless there is a legitimate Sharia reason. Efforts should be made to reconcile between the two families due to the great good inherent in it. It is recommended to seek Allah's help and supplicate to Him to unite hearts.