1) COMPARING "HAZA'AH" ON SHABBOS TO OTHER ACTS OF "SHEVUS"
QUESTION: Rebbi Eliezer in the Mishnah (65b) maintains that when one prepares to offer the Korban Pesach on Shabbos Erev Pesach, he is permitted to perform an action that is prohibited because of Shevus. Rebbi Akiva argues that if one is permitted to perform an act of Shevus on Shabbos for the sake of a Mitzvah, then he should also be permitted to perform Haza'ah.
The Gemara concludes that Rebbi Eliezer agrees with Rebbi Akiva that Haza'ah is not permitted on Shabbos even for the sake of a Mitzvah. The Gemara gives two possible reasons for why it is prohibited. First, a person who is Tamei has no obligation to bring the Korban Pesach, and therefore Haza'ah is not necessary. Furthermore, one may not perform Haza'ah on Shabbos lest he carry the water through Reshus ha'Rabim. Second, Haza'ah is unnecessary because the person does not need to eat the Korban Pesach, but only to offer it in the Beis ha'Mikdash. He may offer the Korban even while he is Tamei by sending it with a Shali'ach.
According to these two reasons for the prohibition of Haza'ah on Shabbos, Rebbi Akiva's comparison between Haza'ah and other acts of Shevus is difficult to understand. Haza'ah is entirely unnecessary, but other acts of Shevus (such as carrying the animal from outside of the Techum to the Azarah) are certainly necessary for offering the Korban. How can Rebbi Akiva prove from Haza'ah that other acts of Shevus for the sake of a Mitzvah on Shabbos are also prohibited?
ANSWER: TOSFOS (DH Katan) explains that the reasons that the Gemara gives for the prohibition of Haza'ah on Shabbos are only Rebbi Eliezer's reasons for why Haza'ah is not compared to carrying the animal. Rebbi Akiva, however, maintains that Haza'ah is prohibited for other reasons (because it is not an essential part of offering the Korban Pesach; see Rashi to 66a and Insights to Pesachim 66:1). Rebbi Akiva's reasoning also prohibits other acts of Shevus, such as carrying the animal. Those acts are also not essential parts of the Korban Pesach, because they could have been done the day before Shabbos.
2) PROOF THAT A "YACHID" WHO IS "TAMEI" DOES NOT NEED "HAZA'AH" ON EREV PESACH
QUESTION: The Gemara says that one who is Tamei (with Tum'as Mes on the seventh day of his purification process, or one who is Tamei with Tum'as Sheretz) has no obligation to have Haza'ah done in order to enable himself to bring the Korban Pesach and eat it that night. Even though he is able to make himself Tahor, he is not obligated to do so. Consequently, Rebbi Eliezer does not permit Haza'ah on Shabbos Erev Pesach.
The Gemara explains that the reasoning of Rebbi Eliezer is based on three steps:
First, the Gemara (end of 69a) says that "Ein Shochtin v'Zorkin Al Tamei Sheretz." One who is Tamei with Tum'as Sheretz (who has the same degree of Tum'ah as a person with Tum'as Mes on the seventh day of his purification process) is not obligated to appoint someone to bring the Korban Pesach on his behalf. Second, once we know that he does not bring the Korban Pesach on Pesach (but he waits until Pesach Sheni to bring it), it follows that if the entire nation is Tamei with Tum'as Sheretz, they are required to bring the Korban Pesach and they do not have to do Haza'ah on themselves (because "Tum'ah Hutrah b'Tzibur"). Third, since the entire nation does not have to do Haza'ah, so, too, an individual who is Tamei with Tum'as Sheretz does not have to do Haza'ah.
Why does the Gemara need these three steps in order to prove that Haza'ah is not necessary for an individual who is Tamei with Tum'as Sheretz? The first step alone suffices, because it states that one who is Tamei with Tum'as Sheretz does not bring the Korban Pesach. That statement alone teaches that Haza'ah will not make him fit to eat the Korban Pesach, and thus there is no basis to perform Haza'ah for him on Shabbos.
(a) RASHI explains that "Ein Shochtin v'Zorkin" means only that before the Tamei person immerses in a Mikvah, a Shali'ach may not perform the Shechitah and Zerikah of the Korban for him. However, perhaps he is obligated to immerse, and after he immerses -- while he is still a Tevul Yom -- a Shali'ach may perform the Shechitah and Zerikah for him and thereby enable him to eat the Korban at night. It is only before his immersion that a Shali'ach may not bring the Korban for him, because circumstances might prevent him from immersing and, consequently, he will not be able to eat the Korban after nightfall. Therefore, the Gemara must prove that he has no obligation to immerse in the first place, even though immersing would permit him to bring the Korban Pesach. The Gemara proves that he has no obligation to immerse from the fact that the Tzibur never needs to perform Haza'ah in order to bring the Korban. (MAHARSHA)
(b) The RAMBAM (Hilchos Korban Pesach 6:6) has a different approach. He explains that "Ein Shochtin v'Zorkin" applies even after the person immersed in a Mikvah. Any person who was Tamei, or a Tevul Yom (who immersed during the day and becomes Tahor at sunset), on the fourteenth of Nisan cannot bring the Korban. Why, then, does the Gemara need to prove from the Tzibur that an individual is not obligated to make himself Tahor? He obviously has no obligation to become Tahor, because that will not make him fit to bring the Korban!
The Rambam explains that the Mishnah here is not discussing the fourteenth of Nisan (Erev Pesach), but rather the thirteenth of Nisan (the day before Erev Pesach). On that day, a person still has time to do Haza'ah and immerse, which will make him fully Tahor by the time Erev Pesach arrives. Therefore, the Gemara finds it necessary to prove that since the Tzibur never has an obligation to do Haza'ah in order to bring the Korban Pesach, an individual also has no obligation to do Haza'ah, even on the thirteenth of Nisan.
3) A "YACHID" DOES MORE THAN A "TZIBUR"
QUESTIONS: The Gemara asserts that there is nothing that an individual (Yachid) does which a Tzibur does not do. The Gemara asks that a Yachid who was Tamei on the fourteenth of Nisan brings a Korban on Pesach Sheni (the fourteenth of Iyar), while a Tzibur does not bring a Korban on Pesach Sheni.
The Gemara answers that this is not considered something that a Yachid does which the Tzibur does not do, because the Tzibur (that was Tamei) brings the Korban Pesach on the fourteenth of Nisan.
RASHI explains that there are times when the Tzibur does not bring a Korban Pesach at all -- neither on Pesach Rishon nor on Pesach Sheni, such as when most of the people are Tamei with Tum'as Zav and the remaining few are Tamei with Tum'as Mes. Since the Tum'ah of a Zav is not Hutrah b'Tzibur, the Tzibur does not bring the Korban Pesach Rishon. They do not bring the Korban Pesach Sheni because there is no Pesach Sheni for a Tzibur.
There are a number of problems with Rashi's explanation.
First, if the Tzibur does not bring a Pesach Sheni even when they did not bring a Pesach Rishon, then the Gemara's question returns. There indeed exists for a Yachid an obligation that does not exist for a Tzibur -- the Korban Pesach Sheni. According to Rashi, how does the Gemara answer this question?
Second, why does Rashi need to mention that there are times when the Tzibur does not bring a Korban Pesach at all? The Gemara says the opposite -- that the Tzibur always brings a Pesach Rishon.
Third, why does Rashi mention that a minority of the Tzibur is Tamei with Tum'as Mes? His example of a Tzibur that cannot bring the Korban Pesach on either day is true even when most of the people are Tamei with Tum'as Zav and the remainder are Tahor!
ANSWER: The simple way to understand the Gemara is that Pesach Sheni is not considered something that the Tzibur does not do, because the only reason the Tzibur does not bring a Pesach Sheni is because they already brought a Korban Pesach on the fourteenth of Nisan. However, the Gemara is actually saying something else.
The Gemara is explaining why the Yachid brings a Pesach Sheni. Only in a case in which the Tzibur brought a Korban Pesach on the fourteenth of Nisan does the Yachid bring a Korban on Pesach Sheni, because in such a case he will not be doing more than the Tzibur. Since the Tzibur brought a Pesach Rishon, the Yachid may bring a Pesach Sheni. (That is, the Gemara is not explaining why the Tzibur does not bring a Pesach Sheni. It is explaining why the Yachid may bring a Pesach Sheni.)
To demonstrate this point, Rashi says that in a case in which the Tzibur does not bring a Korban Pesach Rishon (and thus they bring no Korban Pesach that year, since the Tzibur does not bring a Korban Pesach Sheni), a Yachid also does not bring a Pesach Sheni, because he cannot have more obligations than the Tzibur, as the Gemara teaches.