What is the ruling on the prayer of one who is not comfortable with the imam chosen by the people of the town because he is an innovator and makes mistakes in his prayer and sermons, and does not offer lessons or advice, knowing that the one who chose him does not attend the mosque except on Fridays? And is it permissible for him to pray in his home?

The regular Imam (Imam Ratib) is one who has been appointed by the ruler, his deputy, or the Muslim community. He has precedence in leading the prayer over others present. It is not permissible to abandon praying behind him unless there is something in his prayer that invalidates it. If the Imam performs something in his prayer that invalidates it, then you should advise him. If he does not respond, then it is permissible to pray at home, or to resort to the people of the neighborhood to request his replacement.