1) EATING TERUMAH THAT IS TAMEI
QUESTION: The Gemara explains that the reason the Rabanan made a decree that a person becomes Tamei if he eats food that was Tamei with Sheini l'Tum'ah is because he might put Terumah into his mouth when the Tamei food is there, and the Terumah will become Tamei ("Shlishi"). RASHI (DH u'Pasil Lehu) explains that he will thereby transgress the Mitzvah to preserve the Taharah of Terumah.
Why does Rashi not give the much more obvious concern, that if the Terumah becomes Tamei from the other food inside his mouth, he will be eating Terumah Teme'ah, which is punishable with death?
This question may also be asked on the Gemara itself. The Gemara says that the problem is that he might put Terumah in his mouth and "invalidate it" ("u'Pasil Lehu"; that is, make it Tamei). The Gemara should have said instead that he may put Terumah in his mouth and "eat Terumah which is Tamei" (which is a very severe transgression)!
(a) The MAHARSHA says that indeed both concerns exist, and the Gemara and Rashi simply chose to express one of them.
(b) The RASHASH cites the Mishnah in Terumos (8:2) that states that if one was eating Terumah and it became Tamei while he was eating it, Rebbi Yehoshua says that he must spit it out, and Rebbi Eliezer says that he should swallow it so that he does not transgress the prohibition against wasting Terumah. Why is he permitted to swallow the Terumah Teme'ah if it is forbidden to eat Terumah Teme'ah?
The RI BEN MALKI-TZEDEK (Terumos ibid.) explains that at the moment that he starts chewing on the Terumah, the Terumah is already considered to be consumed. When he subsequently swallows it, it is no longer considered an act of eating Terumah Teme'ah. Since the Gemara here is explaining the opinion of Rebbi Eliezer as well as the opinion of Rebbi Yehoshua, it cannot say that the concern is that one might end up eating Terumah Teme'ah, because -- according to Rebbi Eliezer -- he would not be liable for eating Terumah Teme'ah, since he already put the Terumah in his mouth and started to chew it!
2) TAKING A SHOWER AFTER IMMERSING IN A MIKVAH
QUESTION: The Gemara discusses the decrees that were enacted by the Chachamim during their assembly in the upper chamber of Chananyah ben Chizkiyah. Among them was the decree that one who immerses in a Mikvah and then puts his head and most of his body into drawn water (Mayim She'uvim) is Tamei, and the decree that one who immerses in the Mikvah and then has three Lugin of water poured over him is Tamei. What is the difference between the two decrees?
(a) TOSFOS (13b, DH v'Tahor) explains that the common practice (before the decree) was that immediately after immersing in a Mikvah, one would pour over himself drawn water in order to clean himself. The Rabanan feared that people would begin to think that the drawn water is what makes the person Tahor, and not the Mikvah, and therefore they decreed that not only should one not pour drawn water over himself after immersing, but one should not submerge himself into a pool of drawn water either (and if one does so, he becomes Tamei). Even though people did not normally submerge themselves into drawn water after immersing in the Mikvah, the Rabanan decreed that one who does so is Tamei in order to ensure that people do not mistakenly think that pouring water over oneself also makes one Tahor. This was the first decree, and it applies only on the same day upon which the person immersed himself (i.e. when he is a Tevul Yom).
In a separate enactment, the Rabanan decreed that if it is no longer the same day on which the person immersed himself in a Mikvah, he may submerge himself in drawn water, but he may not pour drawn water over himself. The fact that it is a separate day makes it different in one respect from the original situation about which the Rabanan were concerned and which prompted the first decree, and therefore the Rabanan decreed only that one may not have drawn water poured over him. (A decree that one also may not submerge in drawn water would have been different from the original situation in two ways, and thus it would have been be too far removed from the original situation to warrant making a decree.)
(b) The RAMBAM (Hilchos She'ar Avos ha'Tum'ah 9:2) says that both decrees are essentially the same; both pouring water on oneself and submerging in water make a person Tamei. The only difference is that one decree applies to the day of immersion, and one applies to the day after immersion. The original decree applied only on the day of immersion, but subsequently the Rabanan made a decree for the day after as well.
(This explanation is problematic, because the Gemara expresses the decrees in terms of two different actions -- submerging oneself in drawn water, and pouring drawn water over oneself -- which implies that the two decrees are not identical. It could be that the decrees were phrased with two different actions because that is what the people normally did at the two respective times. That is, immediately after going to the Mikvah, they would pour water over themselves, and then the next day (or the night that followed the day they immersed) they would take a bath.)
3) THE JEWS DID NOT ACCEPT THE GEZEIRAH
QUESTION: Why did the Jews not accept the original decree of Shamai and Hillel who decreed that normal, un-watched hands are Tamei, and they only accepted it later when it was reinstated by the disciples of Shamai and Hillel?
(a) The PNEI YEHOSHUA writes that the people could not accept such a strict measure to burn Terumah on account of a Tum'ah decreed by the Rabanan, since the Torah forbids destroying Terumah if it is Tahor. They accepted the decree only later when the disciples repeated it, for by then they had become accustomed to the notion of burning Terumah that is Tamei with a Tum'ah decreed by the Rabanan.
(b) It could be that the students of Hillel who argued with the students of Shamai and wanted to be Metaher the hands maintained that Hillel never participated in the original decree to make the hands Tamei (for it is not logical to assert that Hillel's students would rule contrary to a decree that their own master instituted). That is why it was not accepted. Later, though, when the students of Hillel saw that they were outnumbered, they conceded to the decree and everyone accepted it. (M. KORNFELD)