What is the ruling on a father greeting the one who assaulted his daughter, causing her breakdown? And is it forbidden to sever ties of kinship with an aunt who falsely accused her brother's daughter, knowing that communication with her causes great psychological pain to her niece?

The assault on your sister by your aunt's husband is an abominable and heinous act. Shunning a disobedient person is permissible, depending on the benefit, and may even be obligatory. However, if he repents, it is not permissible to shun him. Similarly, if your aunt repents from accusing your sister, it is not permissible to shun her, and shunning her in this case would not be for the sake of Allah, but rather out of personal resentment. Severing ties of kinship is a major sin among the greatest of sins. Forgiving a Muslim and having a pure heart towards them are among the best acts of drawing closer to Allah. Your sister is advised to lighten the matter for herself and strive to forget the past.