What is intended by the ten rak'ahs before Fajr in the Sacred Mosque, and are they prayed with the intention of Tarawih, or Qiyam al-Layl, or Tahajjud? Are these prayers Sunnah, and is it permissible to perform them in other mosques starting from the first night of Ramadan?

Tahajjud is prayer at night after sleeping, so there is no contradiction between it and Qiyam al-Layl (night vigil prayer).

Praying in congregation after Tarawih is permissible, whether it is a completion of delayed Tarawih or an independent supererogatory prayer after completing Tarawih and Witr (known as Ta'qeeb).

Scholars differed on the dislike of Ta'qeeb, but many of them hold the view that it is not disliked at all. Sheikh Al-Uthaymeen preferred the view that Ta'qeeb is disliked after finishing Tarawih and Witr because the Sunnah is for Witr to be the last prayer of the night.

Delaying some of the prayer until the end of the night or prolonging the standing in prayer until then is better, especially during the last ten nights of Ramadan.

The matter is broad in these issues, and there is no harm in delaying Witr or some of the Tarawih rak'ahs until the end of the night, or in performing supererogatory prayers after the Witr that was performed earlier in the night.