What is the ruling on praying behind innovators, and if it is not permissible, then should the obligatory prayers be performed at home, especially the Fajr prayer, and what is the foundational evidence for that, and what is the solution if all the surrounding mosques are predominantly adherents of this doctrine?

If the innovation of the imam is not disbelieving (non-kufr), then prayer behind him is valid and sufficient. The Companions prayed behind the Khawarij, and Uthman, may Allah be pleased with him, encouraged praying in congregation in the mosque, even if the imam leading the people was an imam of fitnah (strife). Praying behind a wicked person is better than praying alone. However, if it is possible to pray behind someone who is not an innovator, that is preferable.