Is prayer in the mosque obligatory? And if so, what is incumbent upon a thirteen-year-old son whose mother does not allow him to go to the mosque for 'Isha and Fajr prayers out of fear for him, especially since Fajr prayer is very early?

Congregational prayer is a confirmed Sunnah. The Hanbalis hold that it is an individual obligation (wajib 'ayn). According to the majority of scholars, congregational prayer is not obligatory in the mosque; rather, it is sufficient elsewhere. You must obey your mother if she forbids you from going to the mosque out of fear of harm, and pray in congregation at home if you find someone to pray with you; otherwise, pray alone.