Is it permissible for the wife's family to separate between the spouses and demand divorce against their will, and is it permissible for the wife to disobey her family and refuse their demand for divorce, choosing instead to live with her husband?

If the marriage contract was Islamically valid, then she has become your wife, and her father is not permitted to compel her to seek a divorce without a legitimate Shar'i reason. This is because such an act is considered incitement (takhbib), which is forbidden in Islam. If you have paid the Mahr (dowry), then she is obligated to surrender herself to you, and her guardian is not permitted to detain her. She is also not permitted to obey her father in demanding a divorce; rather, she must reside with you and strive to gain the رضا (رضا) of her parents.