What is the ruling on a sister living in this situation and a daughter being separated from her mother, and is it permissible to abandon the father and cut off communication with him?

A mature, sensible woman has the choice to live with either of her parents, and she cannot be forced to live with one over the other. If she is married, she lives with her husband; otherwise, she may choose either of her parents.

The financial support (nafaqah) of a daughter who has no money or earnings is obligatory upon her father, according to what is customary, until she marries.

It is not permissible for either parent to prevent the other from seeing their child, and the child must be commanded to be dutiful and benevolent towards both parents.

Shunning the father and cutting off communication with him is not permissible and is considered a forbidden act of disobedience (uqūq). For no matter how badly the father may act, he still retains the right of fatherhood.