1) How could the Bnei Betairah have forgotten that a Korban Pesach is brought when Erev Pesach falls on Shabbos? This happens every so often in our days. They lived before the calendar was fixed and declared Rosh Chodesh by witnesses. Certainly it must have happened more than once in their lifetime.
2) The azarah was rinsed by the kohaim on Shabbos against the will of the Chachamim. Who is "the Chachamim?" It turns out to be R. Eliezer. He almost certainly lived one or two generations after the churban. Is the gemara saying that they rinsed out the azara against the will of the chachamim (at least one chacham) of a later generation?! Does that make sense?
3) How did they actually process over 100,000 korbanot in a few hours? Did they also do what we do today, hadacha and melicha with salt besides for shechita and avodat hadam? Did they check the lungs for adhesions as we do or was that a later takana?
4) How was the 1st (and only) korban pesach in Midbar Sinai brought? There were only 3 functioning kohanim after Nadav and Avihu died and 600,000 men over age 20, probably several million Jews at the time.
B'kavod,
Sam Kosofsky
1) A number of commentators ask this question and offer answers. The NIMUKEI HA'GRIV cites the Yerushalmi and the Korban ha'Eidah there, which says that it was a Divine scheme which Hash-m executed in order to have Hillel appointed as the Nasi. Hash-m caused the B'nei Beseirah to forget this Halachah so that Hillel would teach it to them and be appointed as their head.
2) Here it seems that the objection occurred at the time that the Kohanim rinsed the Azarah. When the Gemara says that the "Chachamim" who objected is Rebbi Eliezer, it does not mean that he himself objected, because -- as you wrote -- he lived a generation later. Rather, it means that the Chachamim who held like Rebbi Eliezer's opinion regarding the Halachos of Shabbos objected.
3) They worked very fast! Each Chaburah did the bulk of the work for their own Korban. As far as the Holachah and Zerikah of the blood was concerned, there were a lot of Kohanim who were enlisted for the Avodah that day, as the Mishnah implies (64a, "Ha'Kohanim Omdim Shuros Shuros").
(Since the Korban Pesach was roasted, it did not require Melichah to remove its blood -- the Halachah is that roasting by fire draws out the blood).)
4) The Gemara (64b) implies that in general, there are many people included in one Korban Pesach ("there was no Korban Pesach that had less than 10 people for it"), as the only requirement is that there be enough meat for each person to be Yotzei his obligation of eating the Korban Pesach -- one k'Zayis. If the animals they brought in the Midbar were large, then it could well be that the 600,000 men and their families shared, say, 30,000 Korbanos (20 men per Korban), and there were five hours (after Chatzos, in the afternoon), or 300 minutes, to bring the Korbanos, which means that 100 Korbanos were brought each minute. The only part of the Avodah which required Kohanim was the Kabalah and the Zerikah of the blood. The animals might have been arranged in three moving formations in front of the three Kohanim, such that each Kohen was doing the Kabalah and Zerikah of 33 Korbanos per minute, giving him about 2 seconds per each Korban. If we consider that, realistically, an animal has many, many k'Zeisim, then it is likely that many more people were included in each Korban and thus there were much less Korbanos, and thus the Kohanim were certainly able to handle the load.
But it certainly is an interesting point to consider!
Best wishes,
Yisrael