Is it permissible for a wife to falsely accuse her husband of murder and domestic violence to take his son and money from him, while she is betraying him?

The wife remains a wife unless the husband pronounces a divorce, or writes it, or signs it, provided he intends it and is not coerced.

If the husband becomes certain of his wife's infidelity, he is advised to divorce her. Ahmad indeed recommended that a husband separate from his adulterous wife, as there is no guarantee that she will not corrupt his bed and attribute a child to him who is not his.

As for the custody of the child, his mother has a greater right to his custody until he reaches the age of seven, unless she remarries. However, her disobedience forfeits her right to custody.

One should be wary of residing in lands of disbelief due to the corrupting influences that result from it. The Prophet, peace and blessings be upon him, forbade residing among polytheists.