Was Tarawih prayer not legislated by the Prophet, peace and blessings be upon him, and was it Umar ibn al-Khattab, may Allah be pleased with him, who legislated it? And if that is true, was he permitted to do so?

This statement is incorrect. It has been established that the Prophet, peace and blessings be upon him, prayed Tarawih in congregation during Ramadan for several nights, then stopped going out for fear that it would be made obligatory upon the Ummah. Umar's gathering of the people behind a single Imam occurred after the fear of its obligation had dissipated with the death of the Prophet, peace and blessings be upon him. This was in the presence of the Companions and with their approval, and Umar was a rightly-guided Caliph whose Sunnah we were commanded to follow.