Is it permissible to be led in prayer by an imam who makes mistakes in reciting Al-Fatihah (by extending the "Alif" in "Al-Hamd," by vocalizing the "noon" in "Ar-Rahman" with sukoon, and by adding a "yaa" in "ghayril maghdoobi alayhim"), and who does not complete the bowing, and what is the ruling on past prayers performed behind him?

It is likely that the questioner is suffering from waswasa (whisperings/obsessive thoughts). If so, they must disregard these whisperings and not be led by them to declare the prayer invalid. Not vocalizing the noon in "Ar-Rahman" with a kasra (and instead vocalizing it with a different diacritic) is considered a linguistic error (lahn) that does not invalidate the prayer. As for adding the letter yaa in "ghayril maghdubi," this is doubtful, and the default is that the prayer is sound. Straightening the back in bowing is mustahabb (recommended) and not wajib (obligatory).

However, adding an alif in "Al-Hamdu lillah," if it is indeed true and not an illusion, is a linguistic error that invalidates the prayer because it changes the meaning. Consequently, it is not permissible to pray behind this Imam unless he corrects this error. It is obligatory to repeat the prayer if prayed behind him. As for previous prayers, it is hoped that there is no need to repeat them, because the sounder opinion among the Malikis is that prayer behind someone who makes a linguistic error in Al-Fatiha is not invalidated, even if it changes the meaning. In the future, one must be careful to pray behind an Imam whose imamate is not prohibited, if what was mentioned about the Imam is true and not merely an illusion.