What is the ruling on a wife living separately from her husband in a different residence and completely cutting ties with him due to his persistence in forbidden relationships despite being advised, and does he have any legitimate rights over his wife in this situation?

If what you mentioned about your husband – that he talks to women and receives indecent pictures from them – is true, then he has committed a grave transgression, especially since he prays, and prayer forbids indecency and evil.

We advise the following: 1. Increase supplication for his guidance and repentance. 2. Advise him gently, kindly, wisely, and with good admonition. 3. You must obey him in what is good as long as you are under his marital authority, and your dislike for him does not permit you to withhold his legitimate rights. 4. You are permitted to seek a divorce due to his transgression, or khul‘ (divorce at the wife's instance, by returning the dower or other consideration), but it is not advisable to rush into this before attempting all avenues of reconciliation. 5. There is no harm in moving away from him to a separate dwelling if the husband agrees, but this might increase his corruption. 6. He can be advised to marry another woman if he desires, with the condition of justice.