Is it permissible for a Muslim woman to reside with a Kafir?

It is forbidden for a Muslim woman to be in seclusion with a non-mahram man or to cohabit with him, even if for worship, due to the corruption this causes. Authentic hadiths have explicitly stated this, such as "No man is to be in seclusion with a woman unless a mahram is with her" and "The in-law is death." It is a condition for a woman to be allowed to live with a mahram that the residence be spacious and that neither of them can see the other.

In the case of her living next to a non-mahram man, it is impermissible for facilities to be shared between them (such as the kitchen, bathroom, or hallway), otherwise cohabitation becomes forbidden. Seclusion and cohabitation with a disbeliever are considered even more strictly forbidden, unless the disbeliever is a mahram to the woman.

As for the case of a woman embracing Islam while her husband is a disbeliever: if her conversion occurred before consummation of the marriage, separation takes place. If it occurred after consummation, scholars have differed on the ruling, but the preponderant view is that it hinges on the expiration of the waiting period (iddah); if the husband embraces Islam before the iddah expires, they remain married, otherwise, separation takes place. It is impermissible for a Muslim woman to remain married to a disbeliever.