Which is better: performing the Ramadan congregational prayer at home or individually, and at home or in the mosque? And does praying the Fajr prayer in congregation at home make the worshipper under the protection of Allah, or is that exclusive to the mosque?

Praying Taraweeh at home is better if the intention is to encourage family members to pray it, especially if the nearby mosque performs Taraweeh prayers. However, if the Taraweeh prayer depends on him, then it is better for him to pray it in the mosque. It is permissible to perform supererogatory prayers (Nawafil) in congregation at home. As for obligatory prayers, it is not permissible to stay away from the mosque to perform them in congregation at home except for a valid excuse. And whoever prays Fajr is under the protection of Allah, whether he prays it in the mosque or in another congregation, because the Hadith did not stipipulate congregation or the mosque.