All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
The jurists have stated that the wife is obliged to live with her husband wherever he lives and it is impermissible for her to refrain from moving to live with him without a legitimate excuse, such as if she had stipulated a condition prior to the marriage that he is not to move her from her country. In that case, the husband is obliged to abide by this condition according to the most valid view of the scholars. Please refer to Fatwa 224677 about stipulations in the marriage contract.
If the wife becomes arrogant with her husband and disobeys him, then she is a Naashiz (recalcitrant) wife. The way for the husband to discipline his Naashiz wife is explained in the Islamic prescriptions as underlined in Fatwa 85402. Beating the wife is not the first step in disciplining her if she is Naashiz. It should also be noted that the prescribed beating in this context means non-extreme beating which is not so severe that it causes injury. It is impermissible for the husband to beat his wife in a way that causes her harm.
The most serious thing you mentioned about your wife is that she has stopped praying and has committed adultery and removed her Hijab. Such sins are serious and she must turn back to Allaah in repentance. You should advise her and supplicate Allaah on her behalf. Hopefully, Allaah will guide her to the straight path. Please refer to Fatwa 83244 for more benefit. If she sincerely repents from her sins, then this is the desired objective; otherwise, you are advised to consider divorcing her. It is strongly recommended to divorce such a woman unless the general good of staying married outweighs the harms.
Ibn Qudaamah said about the categories of divorce: “Fourth, a divorce that is recommended, as is the case when the wife neglects her duties with respect to Allaah such as prayer and he (her husband) cannot compel her to observe them.”
If your wife remains as she is, committing those sins, then she does not have rights to custody of the children. A person like her is not to be trusted with the children’s religiosity and moral character. The objective of child custody is serve their best interests. Please refer to Fataawa 90461 and 128584.
Finally, we would like to emphasize the gravity of living in a non-Muslim country and that a Muslim should not settle in a non-Muslim country without necessity or a real need for it. One aspect of this gravity is the man-made law that limits the husband authority with respect to his wife and children, which leads to people miss out on a great deal of their religious interests.
Allaah Knows best.