Is a wife considered divorced merely because her husband asked her to leave the house, or has the divorce not yet occurred? And is it permissible for her to leave the house to visit relatives and friends during her stay at her family's home, given that her husband used to refuse that?

Marital problems are natural, and some spouses face them wisely and succeed in solving them, while others do not. Your sister erred by abandoning her husband; it is not permissible for a wife to abandon her husband. A husband's request for his wife to go to her family's house is not considered a divorce unless he intended divorce. It is not permissible for a wife to leave without her husband's permission, even if she is at her family's house, and your brother has no right to ask your sister to return to her family's house with her clothes unless it is with her husband's permission.