Is it obligatory to make up the missed Sunnah of Fajr, and is it permissible to make it up at home before the Fajr prayer, or after the Fajr prayer in the mosque?

One should take the initiative to make up missed prayers at any time. It is not a condition to make them up in the mosque; rather, it is permissible to do so at home. The Fajr supererogatory prayer (Sunnah) is not obligatory, but it is a Sunnah according to some jurists to make it up absolutely due to its emphasis. Its making up takes precedence over the obligatory prayer (Fard), whether in the mosque or at home.