What is the ruling on congregational prayer for Qiyam al-Layl at different times, such as every Tuesday, or during the first ten days of Dhul-Hijjah? And which of these actions is correct?

There is no doubt that performing the night prayer (Qiyam al-Layl) in congregation during Ramadan is a Sunnah. It is permissible to perform it in congregation outside of Ramadan, provided that it is not taken as a regular habit or an obligatory Sunnah. An Imam who leads people in prayer at different times is closer to adhering to the Sunnah.