Is the imamate of one who recites the opening supplication (Du'a al-Istiftah), then Al-Fatihah, and after "Amin" says: "O Lord of the worlds, O Allah, forgive me and my parents," then recites the supplication "O Allah, distance me from my sins as You have distanced the East from the West" before reciting some verses of the Qur'an, valid?

The Imam's utterance of "O Lord of the Worlds, O Allah, forgive me and my parents" after the Fatihah is not legislated, but it does not invalidate the prayer, nor does it necessitate prostration of forgetfulness (sujud as-sahw). As for the supplication "O Allah, distance me from my sins...", it is a valid opening supplication (dua al-istiftah), but its place is before the Fatihah, not after it. An Imam performing a legislated remembrance in an incorrect place does not invalidate the prayer. If he forgets to do so, there are two narrations regarding prostration of forgetfulness: one states that prostration is not legislated because deliberately omitting it does not invalidate the prayer, and the other states that it is legislated but is recommended, not obligatory. The Imam's prayer is valid, and praying behind him is valid. It is advised to adhere to the remembrances in their correct places.