Is it permissible to pray the Sunnah of Fajr and the Fajr (obligatory) prayer during the prohibited time (after the Fajr prayer has concluded and before sunrise), and would it be considered qada' (a make-up) or ada' (on time), or is it obligatory to wait until after sunrise to perform them as qada'?

The time for the Fajr prayer extends from the emergence of true dawn until sunrise. So, whoever prays Fajr before sunrise, even if by half an hour or less, has performed it within its prescribed time. The proof for this is the saying of the Prophet (peace be upon him): "Whoever catches one rak'ah of Fajr prayer before the sun rises, he has caught Fajr." It is obligatory to hasten to perform the prayer upon waking up, even if that is during a prohibited time, based on the saying of the Prophet (peace be upon him): "If one of you oversleeps or is heedless of the prayer, let him pray it when he remembers it, for Allah says: 'And establish prayer for My remembrance.'"