Why did Moshe instruct Aharon to take specifically a calf as a Chatas?
Rashi: It was to publicize that Hakadosh-Baruch-Hu was pardoning Aharon for making the Golden Calf. 1
Ramban (citing the Sifra) and Moshav Zekenim #1: It was to atone 2 for his participation in the sin of the Golden Calf. 3
Targum Yonasan: It was to prevent the Satan from prosecuting Aharon for his participation in the sin of the Golden Calf. The ram for an Olah was, so that Hashem should recall on his behalf the merit of Yitzchak, whose father bound him like a ram on Har ha'Mori'ah.
Moshav Zekenim #2: It corresponds to Avraham, who took a 'ben Bakar' (Bereishis 18:7) for his guests. 4
Rosh and Da'as Zekenim: Because the Kehunah was taken from [Aharon] through a calf, and through a calf it was returned to him.
See Sifsei Chachamim and Ba'al ha'Turim.
Moshav Zekenim: Even though a Chatas does not atone for Meizid, since Ahron was an Oneis, it was considered like Shogeg.
Ramban: Because when Hashem commanded Moshe in Pikudei, Sh'mos, 10:1 that Aharon should take a bull, the sin of the Eigel had not yet taken place.
Moshav Zekenim #2: By the same token, the ram for an Olah corresponds to [what Avraham offered in place of] Yitzchak; and the Gematriya of "Ayil le'Olah" equals that of 'Ya'akov', and they were Temimim corresponding to Ya'akov "Ish Tam".
Why did Hashem insert the (otherwise superfluous) word "Kach L'cha" Eigel"?
Yoma, 3b: To inform Aharon that he had to pay for his Korbanos out of his own pocket ? and not from public funds. 1
See Torah Temimah, note 3.
Why did Hashem refer to the calf as "Eigel ben Bakar"?
Yerushalmi Nazir, 5a: To teach us that 'Shor' incorporates a calf. 1
See Torah Temimah, note 4.
Where was Moshe commanded to instruct Aharon and the people to bring these Korbanos?
Ramban: The Torah does not record Hashem's instructions to Moshe in this regard 1 as it is self-understood. 2
Even though it does state later in Pasuk 6 - that He did indeed instruct him (Ramban).
Like we find in Sh'mos 16:32 and in Bereishis 31:13. Refer also to Sh'mos 11:1:4:1*.
Why were the Korbanos listed here not mentioned in Tetzaveh Sh'mos, 29:1 together with the Korbanos that were destined to be brought on the occasion of the Milu'im?
Ramban #1 and Moshav Zekenim: Because, whereas the Korbanos listed there were in honor of the Shiv'as Y'mei ha'Milu'im, those listed here on the eighth day were brought by the Kohanim themselves in order to inaugurate them. 1
Ramban #2: The Korbanos here were to atone for the sin of the Golden Calf, 2 which took place after the initial command to build the Mishkan was issued.
