Is it obligatory to repeat the prayer for one who prayed behind an Imam who does not recite Surat Al-Fatihah well, or who doubts his recitation of it?

The default is that the prayer is valid, and it is not nullified except by certainty. If you doubt the occurrence of something that nullifies it, do not pay attention to it, and adhere to the default, which is validity, until nullification is proven with certainty. Not every mistake in Al-Fatihah invalidates the prayer, and based on this, you do not need to repeat it.