Who are "Bnos Tziyon"?
Rashi: They are Banim Metzuyanim (distinguished) via Milah, Tefilin and Tzitzis.
Seforno: They are legions of Hashem.
Malbim (Melitzah): They are physical powers.
Malbim (Mashal): They are esteemed daughters of Tziyon.
Why should they go out?
Malbim (Melitzah): Via this, the Kadosh beloved will finish sending away Bnos Tziyon, lest they cease her connection with her supreme Beloved. She tells them, no one should be with us when we are isolated.
Malbim (Mashal): This is a strategy to enable her to be isolated with her Beloved.
What should they see?
Seforno: You thought that the crown returned to its old state. See the difference between the Mishkan, which was before the Meraglim, and Bayis Rishon, which was afterwards! Even though fire descended also at Chanukas Bayis Rishon, it was greater in the Mishkan, which did not have as much silver, gold and Korbanos. Hashem's cloud was over the Mishkan; once Moshe entered the Ohel Mo'ed, he heard His voice. Since there was sin, and the Teshuvah did not reach Kisei ha'Kavod, like it will in the future - "v'Shavta Ad Hashem Elokecha" (Devarim 4:30). The pardon was not full to [fully] return the crown to its old state.
Malbim (Melitzah): See ha'Melech Shlomo - also the king (Yetzer) has Shalom - there is no longer war between him and the Divine Nefesh!
Who is Shlomo's mother?
Torah Temimah citing Ta'anis 26b: Shlomo is the King that Shalom is His. Imo is Keneses Yisrael 1 . Rashi citing a Midrash - a parable for this is a king who had an only daughter. He loved her so much that he called her Biti ("Shim'i Vas u'Re'i" - Tehilim 41:5), Achosi ("Pischu Li Achosi Rayasi" - below, 5:2) and Imi 2 "Hakshivu Elai Ami u'Le'umi Elai Ha'azinu" (Yeshayah 51:4) - u'Le'umi is written like uL'Imi (missing a Vov).
Seforno: Refer to 3:11:6:3.
Malbim (Melitzah): Physicality and its powers is the mother from which the king (Yetzer ha'Ra) came out.
Perush Rashi Mefurash: Imo is His Umah. Vilna Gaon - the Yerushalmi (Yuma 1:5) says that Hashem sold the Torah. I.e. after Matan Torah, He merely says teachings in the name of Chachamim. The upper beings need the lower, just like a father cannot beget without a mother.
Rinah Shel Torah (ha'Netziv): If a man has many children, and his wife finances and guides them, even the father calls her Em, for she is the mother of the house and his children. Keneses Yisrael enacts and adds good conduct, so it is as if He calls her Imo.
What is "Yom Chasunaso"?
Torah Temimah citing Ta'anis 26b: It is Matan Torah.
Seforno: It is the day of Chanukas ha'Mishkan.
Malbim (Melitzah): It is the wedding of the supreme Beloved with His beloved (the Nefesh).
What crown did she give to him?
Torah Temimah citing Ta'anis 26b: Shlomo is the King that Shalom is His. Imo is Keneses Yisrael. Yom Chasunaso is Matan Torah. (Rashi - then, Yisrael crowned Hashem to be King, and accepted His yoke.)
Rashi citing a Midrash: It is Ohel Mo'ed, which is crowned with colors - Techeles, Argaman and Tola'as Shani.
Seforno: The congregation of the Nesi'im 1 , who were a mother in Yisrael, acquired to Hashem.
Malbim (Melitzah): The body and its powers sacrifice themselves to be totally burned on the Mizbe'ach of love. The body will become spiritual when it hears Mitzvas Elokav. This is the crown of Chasanim - acting according to intellect. Via this, the Chasan mates with the Kalah (the Kadosh Neshamah) to guard her directives and serve her.
The crown was their Korbanos of Chanukas ha'Mishkan. (PF)
What is "Yom Simchas Libo"?
Torah Temimah citing Ta'anis 26b: It is Binyan Beis ha'Mikdash (i.e. the day of Chanukas Bayis Rishon 1 - Seforno).
Rashi: It is the eighth day of the Milu'im, in which the Mishkan was inaugurated in the Midbar.
Maharsha (26b): The Mishnah said that the happiest days for Yisrael were Av 15 and Yom Kipur, and supports this from our verse. The Bartenura says that Chanukas ha'Bayis was on Yom Kipur. What is his source? (Mo'ed Katan 9a says that they celebrated the seven days before Sukos. Perhaps all were days of great Simchah, and Yom Kipur was greatest, because it is also the day of Kaparah - PF.)