Is the Fajr prayer valid when the first adhan is heard, or is it obligatory to wait for the second adhan to perform it along with the Subh prayer? And are the rak'ahs performed after the first adhan counted as part of Qiyam al-Layl (night prayer)?

The time for Qiyam al-Layl (night prayer) extends until the second Fajr (dawn) appears. Qiyam al-Layl can be performed continuously or intermittently. It is best to conclude it with Witr prayer, in accordance with the Prophet's (peace and blessings be upon him) saying: "Make the last of your prayers at night Witr." The first adhan (call to prayer) has no bearing on determining the time for Qiyam al-Layl. As for the Sunnah of Fajr, it begins after the true dawn appears, between the last adhan and the Fajr prayer.