When is the time for the Fajr prayer, and is it permissible to perform it immediately after dawn, while it is still dark? And is the Sunnah of Fajr what is meant by "the two rak'ahs of Fajr" among the twelve regular Sunan prayers?

The scholars have unanimously agreed that the beginning of the time for Fajr prayer is immediately after the appearance of the true dawn (al-fajr al-sadiq), which is the second dawn. So, if one hears the adhan or it becomes clear to him that the dawn has appeared, it is permissible for him to pray at the beginning of the time. This is recommended for those upon whom congregational prayer is not obligatory. As for those upon whom congregational prayer is obligatory, they must pray with the congregation of the mosque. The Sunnah of Fajr is the two rak'ahs before the Fajr prayer itself, and not the Fajr prayer itself.