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1)

(a) 'Achorehem' means the back of a receptacle that is itself carved in such a way that is a mini-receptacle - that it is able to hold water.

(b) The Rabbanan decreed Tum'ah even via the outside of glass utensils, because their similarity to metal vessels, inasmuch as, unlike earthenware ones, they can be repaired when they break.

(a) Vessels made of wood, leather, bone and glass which break, lose their Tum'ah. When they are repaired, they become, once again, eligible to receive Tum'ah, but are not Tamei retroactively.

(b) The Rabbanan glass vessels to metal ones only with regard to Tum'ah d'Oraysa, but when it comes to Tum'ah de'Rabbanan (of which retroactive Tum'ah is one), they gave it the Din of earthenware, to which retroactive Tum'ah does not apply.

(c) They also gave flat vessels made of glass the leniency of flat earthenware vessels (which are not subject to Tum'ah at all). They did this as a reminder that the Tum'ah of glass vessels is only mi'de'Rabbanan, to remind people not to burn Terumah and Kodshim which touch them.

4)

(a) The Tum'ah of metal vessels is d'Oraysa, but Shimon ben Shetach decreed retroactive Tum'ah on metal vessels which broke and were repaired.

(b) Queen Shel Tziyon (alias Alexandra Shalomis, wife of King Yanai) once prepared a banquet for her son. When all her vessels became Tamei, she broke them all and gave them to the metal-smith to repair, in order to be able to use them immediately, and not to have wait seven days. That was when Shimon ben Shetach (her brother) decreed retroactive Tum'ah on all those vessels, and established it as a decree.

(c) The reason of ensuring the continuity of the ashes of the Parah Adumah, is acceptable only according to those who confine the decree to vessels that became Tamei Mes, but not if we extend it to all kinds of Tum'ah, which have no connection with the Eifer ha'Parah?

(d) Rava ascribes the reason for the decree to people who see the owner using his vessels on the same day that they became Tamei, and who are likely to say that Tevilah helps to permit Tamei vessels on the same day that they were Toveled, even before nightfall - and even for Terumah (which requires Ha'arev Shemesh).
If he smashed the vessel completely, then, according to Abaye, we must still suspect that he may leave a large piece intact without a hole the size of a pomegranate, and is still subject to the decree; whereas according to Rava, the vessel will be Tahor when he re-makes it, since everyone can see that the vessel has been re-formed.

(a) For a pipe not to invalidate the Mikveh, it must have been permanently fixed in place, before being carved out in the shape of a pipe.

(b) If the vessels were placed there on purpose, everyone agrees that if the water then flows from into the Mikveh, the Mikveh is Pasul, because it has been filled with drawn water. The initial Machlokes (and ultimate case in which Beis Hillel concede to Beis Shamai, speaks in a case of Shogeg, though we have yet to see exactly what that means).

(c) The Chidush of small vessels is, that we do not say that because of their insignificance they are not sufficiently Chashuv to render the Mikveh Pasul; And large vessels (see Tosfos d.h. 'Echad'), vessels made of plain (unmixed) earth, stone or marble cannot receive Tum'ah, and we may have thought that they cannot render the Mikveh Pasul, either.

(a) According to Rebbi Meir, if the vessels were left lying, not under the pipe, but in the courtyard, then even Beis Shamai agree that, if the rain water then flowed into them, and from them into the Mikveh, the Mikveh is Kasher. Why?
Because, since he did not leave the vessels under the pipe, their is no evidence that he intended the water to flow into the vessels, in which case it will not invalidate the Mikveh.

(b) If he placed the vessels under the pipe (and forgot them there) ...

1... when the sky was overcast, they will all agree that the Mikveh will be Tamei, since, he definitely intended the rain to fill his vessels, and even if he then forgot about them and went away, his original intention does not dissipate.
2. ... when the sky was clear, they all agree that the Mikveh remains Kasher, since there is no indication that he wanted the vessels to be filled.

(c) Beis Shamai and Beis Hillel initially argue over vessels that were placed under the pipe and forgotten there, when the sky was overcast, but then cleared and became overcast again. Beis Hillel hold that since the clouds cleared, he thought that the rain would no longer fall, and his original intention is negated. Whereas according to Beis Shamai, his original intention does not become negated, since the sky became overcast again.

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