I just wanted to clear out something. For mutah, does the girl need permission to leave the house if she’s living with him? Is there the concept of Ataa’at in mutah?

Ruling 2445. A woman in a temporary marriage can leave the house without the permission of her husband. However, if the act of leaving the house violates the right of her husband, then it is unlawful for her to leave the house. And based on recommended precaution, in case the husband’s right is not violated, she should not leave the house without his permission.