Does a wife become legally forbidden to her husband if he forbids her from seeing his female cousin's daughter, but the wife hosts her in their home, or if the husband swears an oath of divorce that his wife shall not leave the house, but she does leave it?

We do not understand what you mean by "My husband is forbidden to his cousin," but it appears that he swore an oath of prohibition. If the husband intended divorce, ẓihār, or an oath by this prohibition, then the prohibition takes effect. As for a conditional oath of divorce, the majority of jurists hold that the divorce occurs when the condition is met. This ruling pertains to prohibition or divorce, and the matter requires knowing the husband's intention to determine what is binding upon him.