Did Umar ibn al-Khattab, may Allah be pleased with him, gather the worshippers for Tarawih prayer, and did he supplicate with a specific supplication after each Taslim?

The one who gathered the Companions behind a single Imam for the Tarawih prayer was Umar ibn al-Khattab, may Allah be pleased with him. As for the tahlil (saying "La ilaha illa Allah"), tasbih (saying "Subhan Allah"), sending blessings upon the Prophet (peace be upon him), and similar forms of remembrance after every two rak'ahs in Tarawih prayer, or after every four, this is a point of scholarly disagreement. Some scholars consider it recommended (mustahabb), some permit it, and others deem it an innovation (bid'ah).

Al-Kasani is of the view that the Imam should sit between two tarawih units for a period equal to one tarawih unit, for tasbih, tahlil, takbir (saying "Allahu Akbar"), sending blessings upon the Prophet (peace be upon him), and supplication, because this has been inherited from the predecessors. Ibn Hajar al-Haytami, however, considers sending blessings upon the Prophet (peace be upon him) specifically at this point an innovation. Ibn al-Haj believes that what people have innovated regarding dhikr after every two tasleemat of the Tarawih prayer and raising voices with it is an innovation. What is encouraged is to follow the predecessors, for all good lies in following the predecessors, and all evil lies in the innovations of those who came after.