What does the Imam of Tarawih do during the last ten nights of Ramadan? Does he limit himself to eight rak'ats and leave the shaf' and witr prayers to be performed alone during the night vigil, or does he pray the shaf' and witr twice?

The Sunnah (definition) for someone who wants to perform night vigil prayer (Qiyam (definition) al-Layl) is not to pray Witr at the beginning of the night. It is preferable to limit oneself to eight or ten rak'ahs, followed by the Witr prayer after the Tahajjud prayer at the end of the night. However, if a Muslim prays Witr at the beginning of the night and then wishes to perform Qiyam at the end of it, there is no impediment, but he should not repeat the Witr.