Why did Moshe need to instruct Aharon to approach the Mizbe'ach? Why did he not simply instruct him to bring his Chatas?
Rashi: Because, ashamed of his sin 1 in the Golden Calf, Aharon was afraid to approach the Mizbe'ach, until Moshe encouraged him to do so, adding 'That is why you were chosen'. 2
Ramban #1 and Moshav Zekenim #1: He was instructing Aharon to approach the north side of the Mizbe'ach in order to prepare the Chatas and the Olah, which required Shechitah in the north. 3
Ramban #2 and Moshav Zekenim #2: He was afraid to approach, for his sin was fixed in his mind, and he saw the form of a calf 4 blocking his Kaparah. So Moshe told him 'Take courage, because Hashem has already accepted your deeds!'
This was Aharon's only sin and it consumed him with guilt (See Ramban).
Moshav Zekenim: You were chosen due to the Chet ha'Egel. We only appoint a leader who has a dubious past, so that, if he will become haughty, we will tell him to step back. Alternatively, it can be interpreted to mean that it was precisely because he was ashamed of sinning that he was chosen to be Kohen Gadol.
Ramban and Moshav Zekenim: He did not specifically say so, since Aharon already knew that Kodshei Kodshim had to be Shechted in the north.
Targum Yonason: He perceived the Mizbe'ach with its corner blocks like a calf. Some say that the Satan created this image.
Why did Moshe say "Vechaper Ba'adcha u'Ve'ad ha'Am" between Aharon's Korban and the Korban of the people?
Ramban and Rosh: To teach him that his Korban must precede that of the Tzibur. 1
Ramban and Rosh: Because, before cleansing others, one needs to first cleanse oneself - 'Yavo Zakai Vi'yechaper al ha'Chayav'.
Why did Yisrael require two Kaparos?
Oznayim la'Torah: One for their own sin and one for causing Aharon to sin, Since a leader only sins ... if the people have done something to cause it. 1 Conseuently, they required an individual Korban to atone for their personal sin and needed to share in Aharon's Korban to atone for having caused him to sin.
See Oznayim la'Torah, who compares this to the Gemara in Bava Kama, 52a 'When the shepherd is angry with the flock, he places a blind goat at its head' and elaborates
Who brought the regular Korbanos on the eighth day of the Mi'lu'im?
Rashi and Ramban: Moshe. Refer to Sh'mos 40:29:2:1.
