What is the ruling on someone who completed reciting Al-Fatihah after the Imam bowed, then completed bowing, then returned to recite Al-Fatihah again under the pretext of not achieving tranquility, so does he repeat the prayer? And what is the ruling on his action? And what is required of him?

Saying "Amin" is not obligatory; rather, it is a recommended act (mustahabb). The prayer is not invalidated if "Amin" is said while bowing for Rukoo‘, even though it is not its proper place. There is no need to repeat the prayer if the mistake was due to ignorance, especially if the pillars of the prayer were fulfilled.