On what date did Yom ha'Shemini fall?
What is it the eighth day of?
Rashi: It was the eighth day of the Milu'im.
The 'Hey' in "ha'Shemini" implies that the day was a special one. In what way was it special?
Rashi (from Seder Olam, Shabbos 87b): Inasmuch as it took ten crowns. 1
Mizrachi (from Shabbos 87b): Ten firsts - The first ... day of creation, ... month of the year, ...day of the inauguration of the Mizbe'ach at the hands of the princes, ... day of Kehunah, ... day of Avodah in the Mishkan, ... day that fire descended from Heaven to burn the Korbanos, ... day that the Kohanim ate Korbanos. ... day that Bamos were prohibited, ... day that the Shechinah descended to the Camp of Yisrael, ... day that the Kohanim blessed Yisrael. Moshav Zekenim - a hint to this is va'Yhi. The letters spell 'Yud Hayu'.
Why did Moshe call the elders of Yisrael?
Rashi: He called them to hear that Aharon was entering the Kehunah Gedolah as per Divine instructions, and not of his own accord. 1
Ramban: Refer to 9:3:1:2.
Da'as Zekenim, Rosh: It was to aggrandize Aharon in front of them.
Preempting Korach's claim in advance. (EC)
Why did the Shechinah not descend until the eighth day?
Moshav Zekenim: This was Midah k'Neged Midah. For seven days, Hashem asked Moshe to go to Mitzrayim, and on day seven, Moshe said "send the one that You will send." Now, for seven days Moshe thought that Kehunah is his 1 , and on day eight, Hashem told him to call Aharon.
Moshe knew that Bigdei Kehunah are for "Aharon u'Vanav"! Perhaps he thought that he himself may always serve without Bigdei Kehunah, just like during the seven days. (PF)
Why did Moshe not call to all of Yisrael?
Moshav Zekenim: He wanted to reveal to Aharon the episode of the Egel, and wanted to do so only in front of modest people. 1
Moshav Zekenim: If so, why did he not do so in private? It was in order that Aharon be shamed in front of them, and get atonement.
QUESTIONS ON RASHI
Rashi writes that it was the eighth day of the Milu'im, Rosh Chodesh Nisan. Perhaps it was Shemini (the eighth) of Nisan!
Moshav Zekenim: It says above "Shiv'as Yamim Yemalei Es Yedchem" (8:33), so Yom ha'Shemini is the eighth of the Milu'im.