Is it valid to perform the Night Vigil Prayer (Qiyam al-Layl) and Witr prayer in the middle of the night, and then pray two rak'ahs a quarter of an hour before Fajr?

The time for Qiyam al-Layl (night vigil prayer) begins after the Isha prayer and ends at the appearance of true dawn (Fajr al-Sadiq). It is best performed in the second half of the night, specifically in the fourth and fifth sixths of it, as Prophet Dawud (peace be upon him) used to do.

As for the two rak'ahs performed a quarter of an hour before dawn, if they are part of Qiyam al-Layl, then they are permissible, with the preference that Witr be the last prayer of the night. If they are the Sunnah (supererogatory prayer) of Fajr, then their time begins with the appearance of true dawn, and it is recommended to perform them early in their time and make them light. To ascertain the time of dawn, one can follow a just mu'adhdhin who is knowledgeable of prayer times.