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1)

(a) In our Mishnah, Beis Shamai say that the Berachah over the day takes precedence over that over wine, whereas in the Beraisa, the Berachah over wine precedes that of the day - i.e. Havdalah.
Then why do they say in the Beraisa that Havdalah (the equivalent to Kidush) comes *after* Yayin?

(b) The author of the Beraisa cannot be Beis Hillel, because then, Besamim should precede Ner.

(c) Neither can the author of the Beraisa be Rebbi Yehudah (in the name of Beis Hillel), because then, Mazon ought to be recited *after* Ner and Besamim (like Beis Hillel say in our Mishnah), and not *before* them, like the Beraisa says.
So the author of the Beraisa has to be Beis Shamai.

(d) The Gemara ultimately differentiates between Kidush and Havdalah. Kidush is recited when Shabbos comes *in*, and it is a Mitzvah to bring in Shabbos as early as possible, so it takes precedence over the wine. Whereas Havdalah, which one recites when Shabbos goes *out*, is said at the end - after the Berachah over wine, because just as it is a Mitzvah to bring in Shabbos as early as possible, so too, is it a Mitzvah to take Shabbos out as late as possible.

(a) The Gemara presumes that having recited the Berachah over the Kos, he probably drinks it immediately, seemingly clashing with Beis Shamai in the above Beraisa, who learns that Birchas ha'Mazon requires a cup of wine?

(b) If he drunk a bit of the wine, he has rendered the remainder of the cup a 'Kos Pagum', thereby disqualifying it from use as a Kos shel Berachah. And besides, if we say that he had sufficient wine to drink and to still leave enough wine for the Kos shel Berachah (a Revi'is), how will we then explain the Lashon of the Beraisa 'Im Ein Sham Ela Kos Echad' ... ?

(c) He did not render the Kos a Kos Pagum, because it speaks when he poured the wine into his hands, from which he then drank. And when the Beraisa says that he only had one Kos, it meant to say that there was not enough wine for *two* cups, but not necessarily that there is not more than *one*.

(d) The Beraisa, which writes that according to Beis Shamai, one first drinks the Kos and then Bensches, disagrees with the previous Beraisa and holds that according to Beis Shamai, there is no obligation to Bensch with a Kos.

(a) The hands are a Sheini le'Tum'ah, and a Sheini cannot make a Shelishi in Chulin (which does not go beyond a Sheini le'Tum'ah) without water (which the hands will make a Rishon). Consequently, without the water, the cup would remain Tahor.

(b) According to Beis Hillel, without the water, the cup could not be Metamei his hands, because a vessel is not Metamei a person.

(c) We are speaking when the back of the cup became Tamei through water, which is only a Tum'ah de'Rabbanan. And, because it is only mi'de'Rabbanan, the Rabbanan decreed that only the outside should become Tamei, but not the inside and all the other sections of the cup (in order that people should remember that its Tum'ah is not d'Oraysa, and will remember not to burn the Terumah that is later contaminated by it. That explains why the inside of the cup is not Metamei the wine inside it.

(d) Beis Shamai forbid the use of a cup whose outside only is Tamei, in case drops of wine spill on to the back of the vessel, and then transmit the Tum'ah of the cup on to his hands. Consequently, Beis Hillel's reason (that the water on his hands may become Tamei through the back of the cup, is not applicable).
Beis Hillel, on the other hand, do not share Beis Shamai's concern about drops spilling on to the back of the cup, because that is uncommon.

4)

(a) We have already answered why the Rabbanan decreed that by Tum'ah de'Rabbanan, only the outside of the vessel becomes Tamei, and not the rest of the vessel, in 3c.

(b) In any event, Beis Hillel claim, one should pour the wine first - because of the principle 'Teikef li'Netilas Yadayim, Se'udah' (meaning that it is better not to interrupt between washing the hands and 'Motzi', by pouring out the Kos).

(a) Beis Shamai are not worried about the table being Metamei the water on the cloth, because, in their opinion, one is not allowed to eat by a table which is Tamei.

(b) We are speaking about a table which is a Sheini le'Tum'ah, and a Sheini cannot make a Shelishi in Chulin without liquids.

(c) Beis Shamai decree eating by a table which is a Sheini by Chulin because of Kohanim who eat Terumah - where there *is* a Shelishi le'Tum'ah.

(d) Beis Hillel are not worried that the Kohanim might become Tamei, because we have a principle 'Kohanim Zerizim Heim' (Kohanim are careful).

7)

(a) Beis Shamai requires that one sweeps before washing Mayim Acharonim, because if one were to wash first, he would probably pour some of the Tamei water on the pieces of bread, rendering them inedible.

(b) Beis Hillel is not concerned about that, because, they maintain that the Shamash will remove the pieces of bread that are larger than a Kezayis, and we are not worried about pieces of bread that are smaller than a Kezayis.

(c) Beis Hillel hold that one is permitted to use a Shamash who is a Talmid-Chacham, and a Talmid-Chacham knows that one has to remove 'crumbs' that are larger than a Kezayis, before Mayim Acharonim. According to Beis Shamai, it is forbidden to use a Shamash who is a Talmid-Chacham, and a Shamash who is not a Talmid-Chacham will not be careful about removing large crumbs.

(d) We learn from Beis Hillel, that it is permitted to destroy crumbs which are smaller than a Kezayis.

(a) The order of priorities according to Beis Hillel is Mazon, 'Ner, Besamim, Havdalah'.

(b) Rebbi Yehudah will learn like this -
Beis Shamai: Mazon, Ner, Besamim, Havdalah.
Beis Hillel: Mazon, Besamim, Ner, Havdalah.

(c) The Halachah is like Beis Hillel, according to the opinion of Rebbi Yehudah.

(a) In the Pesukim from Amos and Yeshayah, the words "Yotzer", "u'Vorei", and "Borei" all refer to the past, negating the Gemara's contention that, according to Beis Shamai, 'Borei' refers to the future, and not to the past.

(b) The Gemara concludes that according to everyone, both 'Bara' and 'Borei' can refer to the past.

(c) According to Beis Hillel, there are many lights in a flame, since each color flame (the red ,the white and the blue) is like a different flame. Whereas Beis Shamai consider it to be one flame.

(a) The candle of a non-Jew is not eligible to be used for Havdalah, because the light did not rest on Shabbos.

(b) Besamim of a non-Jew is not eligible, because we are speaking about Besamim obtained from a group of non-Jews, and most such groups use the Besamim for Avodah Zarah - and Besamim of Avodah Zarah is Asur.

(c) The Seifa of the Beraisa comes to reveal the reasoning of the Reisha, to teach us what we wrote in the previous answer.

(a) A 'Ner she'Lo Shavas' is one which was lit on Shabbos for something which a Jew may not do on Shabbos.

(b) One may use a lamp which was lit before Shabbos and which subsequently burnt for the whole duration of Shabbos.

(c) A light which was lit for someone who is dangerously ill is also permitted, since it is permitted to light such a lamp on Shabbos.