What are the ramifications of "Vehitzilu ha'Eidah"?
Rashi (in Kesuvos, 15a): This teaches us the principle 'Safek Nefashos Lehakeil'. 1
If a Safek occurs in a law concerning the death-sentence, Beis-Din rule leniently.
What is the connection between the Kohen Gadol and the Rotze'ach?
Rashi #1: It is not fitting for the Rotze'ach, who comes to cut short people's lives, to live together with the Kohen Gadol, who comes to bring the Shechinah down into the midst of Yisrael - to cause them to live longer.
Rashi #2 and Targum Yonasan: Because the Kohen Gadol 1 ought to have Davened (in the Kodesh Kodshim on Yom Kipur 2 - Targum Yonasan) that such mishaps 3 should not occur in Yisrael during his tenure (and because he failed to Daven on Yom Kipur, he will die that year - Targum Yonasan). 4
Seforno: Bearing in mind that Galus atones for Shogeg, not all cases of Shogeg are of equal severity - some are more guilty, others less. Consequently, each Rotze'ach requires a time-period in Galus that fits his sin. And Hashem achieves this by arranging the deaths to occur at different stages, 5 some closer to the death of the Kohen Gadol, others, further away, whereas yet others die in Galus before the Kohen Gadol.
Hadar Zekenim, Da'as Zekenim (28): The murderer should not be outside in the Kohen Gadol's lifetime, lest people say 'have you seen this murderer, and the Kohen Gadol does not take vengeance on him?!' It is incumbent on the Kohen to do so
Which Kohen Gadol is the Pasuk referring to?
Ba'al ha'Turim (like Rabanan, Makos 11a): It is referring to a Kohen Gadol who was anointed with the Shemen ha'Mishchah, to one who became Kohen Gadol 1 by wearing the eight garments and to one who stood in for the Kohen Gadol on Yom Kipur when he was Tamei and who stood down when he (the Kohen Gadol) returned. 2
R. Yehudah (Makos 11a): It is referring to a Kohen Gadol who was anointed, or [merely] wore the eight garments, once served in place of the Kohen Gadol, and the Mashu'ach Milchamah (who speaks to the soldiers going to war).
Since when did the murderer anoint the Kohen Gadol?
Rashi #1: It is a 'short Pasuk', which is missing a word; it means 'Asher Mashach ha'Moshech' (the one who anointed him).
Rashi #2 (citing Makos, 11b): Because if the Kohen Gadol dies and they appointed a replacement before the conclusion of the murderer's judgment in Beis-Din, he may only leave the Ir Miklat when the second Kohen Gadol dies (and the Pasuk is simply referring to the Kohen Gadol who was anointed in his time