Why did Aharon now enter the Kodesh Kodshim for the second time?
Rashi and Ramban: To take out the Kaf u'Machtah (the bowl and the pan) 1 which he had brought in earlier with the Ketores.
It is otherwise inconceivable that he should enter the Heichal merely to remove his clothes (See Ramban DH 'u've'Emes').
How do we reconcile the order of the Avodah as it stands with the tradition that on Yom Kipur, the Kohen Gadol Toveled five times and performed ten Tevilos of hands and feet?
Rashi and Ramban (citing Yoma, 32b): We do this by inverting the Pesukim - by inserting the Olah of Aharon and that of the people (in Pasuk 24) here, so that Aharon had to Tovel, change and wash his hands and feet before bringing the Olos 1 and afterwards, before he finally entered to take out the Kaf and Machtah.
And some of the Musafin (Rashi).
Why does the Pasuk invert the order of events?
Ramban: Because it is currently concerned with the Avodos ha'Yom, 1 which it completes before continuing with the regular Avodah.
What was the purpose of the fourth Tevilah (mentioned in this Pasuk)? which Avodah followed it?
Rashi and Ramban #1 (citing the Toras Kohanim): Following the fourth Tevilah and washing of the hands and feet, Aharon brought the rest of the Musafin, the Tamid shel bein ha'Arbayim and the daily afternoon Ketores 1 wearing the Bigdei Zahav.
Ramban #2 (citing Yerushalmi): Aharon only took out the Kaf and the Machtah after the Tamid shel bein ha'Arbayim, 2 leaving only the burning of the Ketores, the Chavitei Kohen Gadol, the Nesachim and the Neiros for after the fourth Tevilah. 3
Seforno: He required Tevilah after leaning his hands on the head of the Sa'ir ha'Mishtale'ach.
Aharon brought the Olos after his Chatas, since a Chatas always precedes an Olah, but why did he bring them after the Sa'ir la'Azazel?
What are the ramifications of "Vehinicham Sham", mentioned in connection with the four white Begadim that Aharon wore on Yom Kipur?
Rashi and Seforno: It means that, after removing the Kaf u'Machtah, he was to change into the eight Begadim of a Kohen Gadol, to leave the white ones there 1 , never to be used again, 2 not by a Kohen Hedyot during the year nor by Aharon on the following Yom Kipur.