Is it permissible to pray behind an imam who prolongs the dammah in "na'budu" in Al-Fatiha, turning it into a waw? And does the ruling differ in a silent prayer? And is it obligatory to repeat the prayers missed behind him if my prayer is not valid with him?

It is not valid for someone who recites Al-Fatihah proficiently to be led in prayer by someone who makes a grammatical error (laḥn) in it that changes the meaning and invalidates their prayer. However, a grammatical error that does not change the meaning, such as prolonging a vowel sound, does not invalidate the prayer of either the Imam or the worshiper, because it does not affect the meaning.