Is it permissible for the husband to practice polygyny despite the refusal of his wife and mother, and the absence of a dedicated home for them, and what should the wife do, and is he obliged to obey his mother in not marrying?

The Sharia permits polygyny for men under two conditions: the ability to provide for the wives and bear the costs of marriage, and fairness among wives in terms of financial support and division of time. The first wife's jealousy is a natural human emotion for which she is not blameworthy, as long as she remains within the bounds of Sharia. As for the mother's anger regarding her son's second marriage, it is misplaced, and she is not permitted to abandon him. The son should try to appease his mother and be patient with her, unless he fears extreme hardship for himself.