Is the Iqamah obligatory for prayer when performing congregational prayer at home, especially if the Imam makes many mistakes or is very fast in prayer in nearby mosques? And does the one who issued a fatwa that the Iqamah is not obligatory bear the sin of leaving it?

First: The adhan (call to prayer) and iqamah (second call to prayer) are obligatory for the prayer, but are not part of it; that is, they are external to it. The evidence for their obligation is the command for them and the Prophet's consistent practice of them. They are a communal obligation (fard kifayah) for both residents and travelers. If a Muslim prays without them, their prayer is valid. There is no blame on someone who makes a mistake in their speech out of ignorance, but it is not permissible for someone to speak about something whose correctness they do not know.

Second: Falling behind in congregational prayer in mosques is not permissible except with a valid excuse. If the imam makes a mistake in the Fatihah that alters its meaning, then it is permissible to fall behind. As for other mistakes or speed that does not invalidate the obligatory acts, his prayer is valid, and any deficiency can be compensated for by supererogatory prayers. If there is another nearby mosque whose imam does not misrecite, then it is obligatory to go to it. The mosque that is immediately adjacent is not a condition, as long as one can hear the call.