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1)

(a) The Chachamim hold that, 'Mei'es Le'es' and 'mi'Pekidah li'Pekidah' are only Metamei the man who was intimate with her for one day, and not seven. According to Rebbi Akiva, he is Tamei for the full seven days, whereas Rebbi Yehudah, son of Rebbi Yochanan ben Zakai maintains that he is even permitted to do the Avodah in the Beis Hamikdash.

(b) Rebbi Yehudah holds like Shamai, that 'Kol ha'Nashim Dayan Sha'atan'.

(c) Nor is he Tamei Keri, because it speaks in a case when the man did not finish his Bi'ah, where there was no Keri.

(a) If we hold like Rav, in whose opinion the Chachamim agree that Kesamim are Metamei the Bo'eil retroactively, then the author of our Mishnah will have to be Rebbi Meir, who is Machmir by Kesamim.

(b) The Chidush of the Mishnah according to Shmuel - who learns that even when the Chachamim agree with Rebbi Akiva that the Kesamim are Metamei the Bo'eil, that is only in the future, but not to be Metamei him retroactively - is that they are more stringent by Kesamim in this regard than they are regarding the Bo'eil retroactively.

(c) Why is that?
Because by Kesamim, there is something real to decree on, whereas by Mei'es Le'es, there was nothing during the Mei'es Le'es on which to decree.

4)

(a) According to Rav Huna, when a woman's fixed Veses has not yet arrived, then we can certainly rely on her Chezkas Taharah, even more than on a woman who has no Veses.
On the other hand, we can rely on the Chezkas Taharah of a woman who has no veses at all, more than we can on a woman who has a Veses which has arrived.
Practically speaking, we rely on a woman's Chezkas Taharah in all cases except for a woman who has a Veses and that Veses has arrived.

(b) According to Raba bar bar Chanah, even if a woman's Veses has arrived, she is still permitted, because, in his opinion, Vestos are de'Rabanan (min ha'Torah we do not say 'Orach bi'Zenamah Hu Ba') . Rav Huna holds that Vestos are d'Oraysa.

(c) Our Mishnah is speaking about a woman who has a Veses, but her Veses has not yet arrived.

5) According to Rav Ashi's version of Rav Huna's statement, our Mishnah is speaking about a woman who has no Veses for days, only for days together with jumping. Consequently, the husband can rely on the fact that his wife did not jump and is therefore Tehorah, but a woman who has a Veses for days only, is forbidden to her husband as soon as her Veses falls due, because Vestos are d'Oraysa.

6)

(a) Yes! Rebbi Yochanan permits the husband of even a young girl who is still embarrassed to go to Mikveh before her husband asks her, to rely on his own reckoning and to be intimate with her, without asking her, because he is speaking about a woman who may have seen , but have not. Consequently, we rely on two Sefeikos: maybe she did not see blood, and, even if she did, maybe she went to Mikveh.

(b) The case of a Chaver who died, leaving a store of grain is either considered a Vaday and Vaday - because it is Vaday that a Chaver (a Talmid Chacham who is particular about Taharah) will not leave his store un'Ma'asered, even for one day.
Or it is considered a Safek and Safek, because even if we are not sure that the Chaver Ma'asered his store of grain, but who said that it was ever Chayav to be Ma'asered? It is possible that he took the grain into his house before winnowing, which exempts the grain from being Ma'asered (min ha'Torah) - and which practically, he may feed his animals, and from which he may himself eat casually.

(a) The maidservant gave birth to a Safek stillborn baby in a pit (we will soon see the two possible cases), and the Kohen went to ascertain whether it was indeed a baby, in which case the mother would have been Temei'ah Leidah. And if it did turn out to be a baby, he also had to ascertain whether it was a boy or a girl (so that the mother would know how many days Tum'ah and Taharah to sit). The Sha'aleh now was whether the Kohen became Tamei be'Ohel ha'Meis or not, when he peeped into the pit. The Chachamim ruled that he was Tahor, apparently because Safek Motzi Miydei Vaday.

(b) This case too, was not a good case of Safeik Motzi Miydei Vaday, because the reason the Rabbanan declared the Kohen Tahor was because, either it was case of Safeik and Safeik - a Safeik Nefel, and even if it was a Nefel, perhaps a weasel came and ate it or dragged it away, so that the kohen was not Ma'ahil on Tum'ah. Or because it was a Vaday and Vaday, since, even if she definitely gave birth to a baby, because there were certainly weasels and Bardelos (snakes or small creatures of prey similar to easels) there, we can be sure that they will have eaten or dragged the baby away.

(c) The Kohen was an Am ha'Aretz, who was not too concerned about the Dinim of Kehunah.

(d) The kohen might have just been a Posek coming to teach the mainservant what she had to do.
Or he could have the master of the household where the maidservand worked, and he simply wanted to know whether she was Temei'ah to keep his Taharos away from her.
Or he may have even just wanted to know whether she was Temei'ah Leidah, in which case, she would be obligated to bring a Korban, the skin of which goes to the Kohanim.
Rashi discards the last explanation, because there was so little in it for him. The skin of one animal shared among a whole group of hundreds of even thousands of Kohanim (who belonged to the Beis-Av serving that particular day in the Beis-Hamikdash).