Does the deputy imam's re-recitation of Al-Fatiha after the original imam has recited it affect the validity of his prayer and the prayer of the congregants, and what are the rulings for deputizing an imam in prayer?

The prayer is valid in all circumstances. What the imam did falls under the legal ruling for those who obligate the congregant to recite Al-Fatihah, which is the established fatwa. So, if the imam designates someone else to lead, the prayer is valid, and the designated leader completes the prayer. He may begin reciting Al-Fatihah from the beginning, or he may start reciting from where the first imam finished, provided the designated leader recites what the first imam had already recited. As for a silent prayer, the designated leader begins the recitation from the beginning, even if the first imam had stood for the duration of reciting Al-Fatihah.