What is "Ya'iv"?
Rav Sadya Gaon, Rashi: It is to darken. Rashi - this is like "veha'Shamayim Hiskadru Avim" (Melachim I, 18:45). Rid - it is past tense written in future tense.
Ibn Ezra: It is raising until Av (clouds).
R. Avigdor: People that it was proper that salvation come via them, You pushed them off like a cloud.
Palgei Mayim: It is future tense. The Shechinah is with us in Galus, for assurance that we will be redeemed. Rashi (Shemos 15:1) explained that it says "Az Yashir" (future), for Moshe put to his heart to sing. Also here, the Megilah comes to teach Hashem's Chesed with us amidst the afflictions (He put to heart to assure us of redemption).
What is "Bas Tziyon"?
Rav Sadya Gaon: The congregation of Tziyon.
What is the significance of "Hishlich mi'Shamayim Eretz"?
Eichah Zuta (1:7): Yirmeyah's uncle Chanam'el used Hashem's name to raise Yerushalayim in the air [lest the enemy destroy it]. Hashem 'kicked' it back down.
Rashi: After He raised them to Shamayim, He cast them to the ground at once. Ibn Ezra - He raised Tziyon only in His anger, in order to cast them down from a high place.
Hadar Zekenim (Shemos 3:4): It does not say la'Aretz, rather, "Eretz" - He cast down the brick that He always looked at to recall the servitude of Yisrael.
What is "Tif'eres Yisrael"?
Ri Kara: It is "Kol Ne'os Yakov
What is "Hadom Raglav"?
Rav Sadya Gaon: The dwelling of His Shechinah. Palgei Mayim - for the good of Yisrael, He did not recall that it is Hadom Raglav [and sent it to Galus with Yisrael].
Rashi: His footstool - the Beis ha'Mikdash. Rid - "v'Shulav Mele'im Es ha'Heichal" (Yeshayah 6:1). Ri Kara - Targum Yonasan of "veha'Aretz Hadom Raglai" (Yeshayah 66:1) is Beis Mikdashi 1 . R. Avigdor - Yisrael ascend there with their feet, for Pnei Hashem to see them there.
Ri Kara: It is His world. The land and all its inhabitants are called Hadom Raglav - "ha'Shamayim Kis'i veha'Aretz Hadom Raglai" (Yeshayah 66:1).
R. Avigdor citing Eichah Rabah 2:3: Hadom is written without a Vov, like ha'Dam - He did not recall the blood between the legs of the [one who circumcised himself when he was] old. Avraham was 99 when he was circumcised (Bereishis 17:24).
R. Avigdor: He did not remember the blood of Milah. It says Zar'acha six times in Parashas Milah, therefore the Vov is omitted (He did not recall its merit for Avraham's seed.)
Our text says 'v'Ar'a Kevash Kadamai.'