Are the two rak'ahs of Fajr and Witr counted among the twelve rak'ahs of the regular Sunnah prayers, or are the regular Sunnah prayers performed in full in addition to them? And is it permissible to perform a supererogatory prayer between the first Adhan and the second Adhan for the Fajr prayer?

The intended meaning of the twelve rak'ahs mentioned in the hadiths refers to the regular sunnah prayers, which are: four before Dhuhr, two after it, two after Maghrib, two after Isha, and two before Fajr prayer.

As for Shafa' and Witr, they are not part of these. There is no objection to performing supererogatory prayers after the first adhan (before Fajr) for one who has not prayed Witr. And for one who has prayed Witr, he may pray as much as he is able, and he is not required to pray a new Witr.