Are the prayers performed by a husband, while wearing clothes with a suspected impure stain whose appearance date he does not know, considered valid, or must they be repeated?

The impurity of a spot is not ruled upon by mere doubt. If one is certain that it is an impurity but does not know when it occurred, their prayer is valid. Likewise, if one is certain that it occurred before the prayer but was unaware of its existence, their prayer is valid, and they are not required to repeat it. Ibn Taymiyyah said: "If one prays with impurity on their body or clothes and does not become aware of it until after the prayer, they are not obligated to repeat it, according to the more correct of the two scholarly opinions."