Is تحية المسجد (the salutation of the mosque) obligatory upon the Imam and the congregants if they pray the two Rak'ahs of Fajr in their homes and then go to the mosque to perform the Fajr prayer?

If they enter the mosque, they should pray تحية المسجد (two rak'ahs upon entering the mosque) before sitting. The Imam may order the establishment of the Fajr prayer, in which case تحية المسجد is not needed because the obligatory prayer suffices for it. The same applies to the worshiper (ma'mum) if he enters and the prayer has already been established.