What is "Sha'ar ha'Tavech"?
Rashi (from Yerushalmi Eruvin 5:1): It is the middle gate, the gate of the Azarah - Sha'ar Nikanor 1 , in the east [of the Azarah]. It is between the gate of Ezras Nashim and the gate of the Heichal. It is where they are Mechatech (decide) Halachos. About this it says "Mekom ha'Mishpat Shamah ha'Resha" (Koheles 3:16).
Radak: It is the gate of the middle wall. It is proper to say that it was a gate of Yerushalayim 2 , for [Sarei ha'Kasdim] stayed there after they breached [the outer wall of] the city - "u'Va'u v'Nasnu Ish Kis'o Pesach Sha'arei Yerushalayim" (1:15).
I.e. in Bayis Sheni it was called so. (PF) Radak - it has six other names. Refer to Melachim II, 11:6:1:1*.
I.e. a gate of the middle of the three walls around Yerushalayim. (PF)
What is the significance of the officers camping in Sha'ar ha'Tavech?
Rashi: This fulfilled "u'Va'u v'Nasnu Ish Kis'o Pesach Sha'arei Yerushalayim 1 " (1:15).
Malbim: They came inside the [outer] wall. Malbim (1, citing Mahari) - even though the wall was breached in Tamuz, they did not enter until the seventh of Av. Yisrael fought them until then, until hunger pressed them.
Rashi explained that Sha'ar ha'Tavech is a gate of the Azarah. Radak learned from the verse that Rashi cites that it is not a gate of the Azarah, rather, of Yerushalayim!
Why are Nergal and Sar'etzer repeated in this verse?
Radak: Perhaps there were others with these same names.