Is it true that forgetting the supplication before sexual intercourse allows Satan to see one's private parts? And is it obligatory to recite the supplication only when caressing one's wife and seeing her private parts?

It is sunnah for one who wishes to engage in intercourse with his wife to say: "In the name of Allah. O Allah, keep Satan away from us, and keep Satan away from what You رزقنا (provide for us)." If a child is destined between them, Satan will never harm it. It has been narrated from some of the Salaf that Satan participates in intercourse if the ذكر (remembrance of Allah) is forgotten, but the hadith does not mention that. It is also prescribed to remember Allah to seek protection from the jinn when uncovering one's nakedness, such as when entering the lavatory, by saying: "In the name of Allah."