Why did Moshe need to instruct Aharon to approach the Mizbe'ach?
Ramban #1, 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, 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. Moshe told him 'do not be so humble; Hashem already accepted your deeds.' Some say that the Satan created this image.
This was Aharon's only sin and it consumed him with guilt (See Ramban).
Moshav Zekenim: You were chosen due to Chet ha'Egel. We appoint a leader only if he has an ignominious past, so if he will become haughty, they will tell him to step back. This can be interpreted to mean that it was precisely because he was ashamed of sinning that he was chosen to be Kohen Gadol. (EC)
Ramban, Moshav Zekenim: He did not specifically say so, since Aharon already knew that Kodshei Kodshim had to be slaughtered in the north, and he would understand it himself.
Targum Yonason: He saw the Mizbe'ach, with its corner blocks, appear like a calf.
Why did Moshe say "v'Chaper Ba'adcha u'Ve'ad ha'Am" between Aharon's Korban and the Am's Korban?
Rosh: He teaches why his Korban comes before the Tzibur's - before cleansing others, one needs to first cleanse oneself. 1
'Ramban, Rosh: Yavo Zakai va'Yechaper al ha'Chayav'.
Who brought the regular Korbanos on the eighth day of the Mi'lu'im?
Rashi and Ramban: Moshe. Refer to Shemos 40:29:2:1.