Does he miss out on the reward of the night vigil (Qiyam al-Layl) if he leaves after completing the entire Tarawih prayer and before the Imam leaves?

There is no harm for the worshipper in leaving the mosque before the Imam if he has completed his prayer with him. The meaning of the Prophet, peace be upon him, saying: "Whoever stands with the Imam until he departs, it will be recorded for him as if he prayed the whole night," refers to the Imam concluding the prayer, not his physical departure from the mosque. The practice of Muslims throughout the ages has been in accordance with this, and scholars have used this hadith (definition) as evidence for the desirability of remaining with the Imam until the prayer is finished.