What is the correct way to perform the Night Vigil (Qiyam al-Layl) prayer and the Witr prayer, and is it better to pray Witr after the Night Vigil or with the Imam during Tarawih prayer?

Both are correct, whether the two rak'ahs of Qiyam al-Layl are prayed before or after the two rak'ahs of Shafa'. It is sunnah to recite Surahs Al-A'la and Al-Kafirun in the last two rak'ahs before Witr, as they are part of Witr. It is better for one who prays Tarawih with the Imam to complete the prayer with him so that he may be credited with a night of worship. If he wishes to pray at home, he should pray Witr with the Imam, then stand and pray one rak'ah, then pray as much as he wishes, and conclude with Witr. It is better for a woman to pray in her home.