Were public baths present in the time of Jacob (peace be upon him), and could one of Joseph’s brothers have been conceived from a sperm cell picked up from these baths, based on the hadith "None of you should bathe in standing water while he is in a state of janabah (major ritual impurity)"?

The questioner should not be preoccupied with a question that involves slandering the lineage of the prophets' sons or denying the Qur’an. He should beware of satanic whispers. The Qur’an states that Joseph's brothers were the sons of Jacob, and whoever doubts this contradicts the Qur’an. The prohibition of bathing in stagnant water is because it becomes foul with frequent use, not because it becomes musta‘mal (used water, unfit for ritual purification). Therefore, beware of speaking about Allah and His Messenger without knowledge.