Is the marriage of a woman to a man who does not pray, mistreats her, and frequently utters divorce pronouncements, while she is forbidden from hanging up the phone on him, with his threat of taking their daughter from her, considered a valid marriage according to Islamic law?

If the husband abandons prayer and drinks alcohol, he is committing grave sins that necessitate advising him, warning him of God, and supplicating for his guidance. If he does not repent, then divorce is the correct course of action. If the husband pronounces divorce explicitly or implicitly with the intention of divorce, his divorce takes effect. As for declaring the wife unlawful (تحريم الزوجة), it depends on his intention (divorce, ẓihār, or oath). These matters require detailed inquiry from the Sharia court to determine the status of the divorce and custody. The divorced woman must not leave the marital home until her iddah (waiting period) ends, and the husband is not permitted to expel her.