1)

(a)According to the Gemara's first contention (Rabah), Chazal forbade Toveling utensils ...

1. ... on Shabbos - because one may come to carry the utensils into the street (on one's way to the Mikvah).

2. ... on Yom-Tov - because otherwise, one may come to Tovel them on Shabbos, too.

(b)They forbade even someone who has a Mikvah on his property (and does not need to carry his utensils) to Tovel them - because of someone who hasn't.

(c)Even though Chazal decreed one case because of the other (as we have just seen), they did not forbid Hashakah because of Tevilah - because it is only someone who does not have drinking water who makes Hashakah, and such a person is usually very careful not to allow his water to become Tamei. Consequently, Hashakah is rare, and Chazal tend not to issue decrees in cases that are rare.

2)

(a)The Beraisa permits Toveling a Tamei bucket by using it to draw water from a well. This differs radically from Toveling Tamei water in a Tamei vessel, which our Mishnah forbids - inasmuch as there, the purpose of the Tevilah is clearly in order to Tovel the water or the vessel; whereas here, by the bucket, who is to know that he is not lowering the bucket into the well in order to draw water?!

(b)Nor do we decree Tevilah using this method, for fear that one may come to Tovel the bucket directly - because, now that Chazal required him to use this method, he will certainly remember that Toveling directly on Yom-Tov is prohibited.

(c)Despite the fact that one is not permitted to Tovel a vessel that became Tamei before Yom-Tov, they allowed one to Tovel one that became Tamei on Yom-Tov - because Tum'ah on Yom-Tov is unusual, and we have already learned above, that Chazal do not generally decree an Isur on something which is unusual.

(d)True, a Toveled vessel requires Ha'arev Shemesh before it becomes permitted - but that is only as regards Terumah; as far as Chulin is concerned, one may use the vessel immediately, so he can always use the Toveled vessel for Chulin.

3)

(a)A vessel that became Tamei through a 'Vlad ha'Tum'ah' (i.e. a Rishon) - may be Toveled on Yom-Tov.

(b)Chazal did not decree a vessel that became Tamei through a Vlad ha'Tum'ah because of one that became Tamei through an Av - because its Tum'ah (which is only mid'Rabanan - since, min ha'Torah, a vessel can only become Tamei through contact with an Av ha'Tum'ah) affects only Kohanim (seeing as a Sheni l'Tum'ah can render Terumah a Shelishi, but not Chulin), and 'Kohanim Zerizin Hem'.

(c)'Kohanim Zerizin Hem' (in this context) means either that - the Kohanim will take care not to permit Toveling vessels that received Tum'ah from an Av ha'Tum'ah; or that they are careful to keep their vessels away from Tum'ah, in which case it is unusual for this to happen, and Chazal did not decree.

(d)They also permitted a Nidah who has no spare clothes, to 'cheat' and Tovel on Yom-Tov in her clothes, even if, at the same time, she Tovels her clothes. They did not decree a Nidah when she has no spare clothes for fear that when she has, she may Tovel them outright - because, since they only permitted her to Tovel in her clothes, she will certainly remember that Toveling clothes on Yom-Tov in a conventional manner is forbidden.

4)

(a)Rav Yosef explains the reason for the prohibition of Toveling vessels (which means 'clothes' in his opinion) on Shabbos, because one may come to squeeze out one's clothes. According to Rav Bibi - it is because we are afraid that if Toveling vessels on Yom-Tov were to be permitted, one may take advantage of this, and postpone the Tevilah of vessels that need Tevilah, for Yom-Tov. Meanwhile, he may forget and use them for Terumah.

(b)According to Rava, the reason for the prohibition of Toveling vessels is because it resembles making a vessel (since effectively, one is making the vessel fit for use). Nevertheless, Beis Hillel permit the Tevilah of a person - because to the onlooker, the person may well have entered the water to cool down (and we have already learned that, any prohibition that is based on Mar'is ha'Ayin, is only prohibited if it is clear that what the person is doing is Asur, but not if it can be translated as something that is permitted).

(c)This does not mean that he is forbidden to Tovel in dirty water, in which one does not usually bathe - because even there, one might come home on a hot day and enter the water to cool down.

5)

(a)It is not even forbidden to Tovel in dirty water during the winter months - because sometimes one comes home from the fields dirty (from the mud), and enters the water, even if it is dirty, in order to clean it off.

(b)The reason that Tevilah is permitted even on Yom Kippur (when it is forbidden to enter the water to cool oneself, and it is evident that he is only doing so in order to Tovel) is because of 'Ho'il' - whatever is permitted (Melachah-wise) on Shabbos, is certainly permitted on Yom-Kippur.

6)

(a)The Mishnah in Shabbos permits a man with tooth-ache to dip his food in vinegar on Shabbos and eat it, but not to sip it directly - the Beraisa writes that one may not sip it and spit it out (because then it was obvious taken as a cure), but that one may sip it and swallow it (when it looks as if he is just drinking it).

(b)Abaye establishes our Mishnah when he sips it and spits it out (which conforms with the Beraisa). Rava reconciles the Mishnah with the Beraisa, even if the Mishnah is speaking about swallowing the vinegar - which is also forbidden after the meal (when it is obvious that he is doing it purely as a cure); whereas the Beraisa, which permits swallowing it, speaks before the meal (when it is feasible to drink vinegar, because it might help develop one's appetite.)

(c)But this appears to contradict what Rava said earlier (in 5d.) - where he applied the principle of 'Ho'il' (from Shabbos to Yom Kippur). Why do we not say here too, that whatever is permitted before the meal, will also be permitted afterwards - because of 'Ho'il'!?

(d)We answer that Rava retracted from one of his statements. This can only be from the latter one (regarding the distinction between drinking vinegar before the meal and after it) - because otherwise, he will clash with a Beraisa, which explicitly writes that, whoever is Chayav Tevilah may Tovel normally, both on Tish'ah b'Av and on Yom-Kippur.

7)

(a)In a Beraisa, Rebbi says that one may not Tovel a vessel together with the water that it contains, and one may not make Hashakah, either. The Rabanan hold that - both are permitted.

(b)In that case, we ask - who is the author of our Mishnah, which permits Hashakah, but forbids Tevilah?

(c)We establish our Mishnah like Rebbi - in which case, it speaks about Yom-Tov, when Hashakah is permitted. Rebbi's first statement too, forbidding Tevilah (but permitting Hashakah) - speaks on Yom-Tov (conforming with the Mishnah). His second statement however, forbidding Hashakah, speaks on Shabbos, when Rebbi agrees that Hashakah is forbidden, too.

(d)Alternatively, we establish our Mishnah like the Rabanan, in which case, our Mishnah speaks about Shabbos - when according to the Rabanan, Hashakah is permitted, but Tevilah is forbidden. Their opening statement (like Rebbi's) refers to Yom-Tov, when they permit even Tevilah; And it is their second statement which (like Rebbi's) speaks by Shabbos, where they permit Hashakah, but forbid Tevilah, that coincides with our Mishnah.