Siman 490, Se'if 9

Shulchan Aruch, Siman 490, end of se'if 9
"It is customary to say Shir Hashirim on the Shabbos of Chol Hamoed, and if Shabbos falls on the last day of Yom Tov, it is said on that Shabbos. The same rule applies on Sukkos with Koheles, and it is customary to say Rus on Shavuos. The people have adopted the practice not to recite a berachah over them, neither 'al mikra megillah' nor 'al mikra kesuvim.'"
(Rema)

What we are not dealing with here: The beginning of se'if 9 discusses a Shabbos that falls during Chol Hamoed, including the Rema's ruling that Pesach is not mentioned in the berachah over the haftarah, neither in the middle nor in the closing. In a year with no Shabbos during Chol Hamoed, this has no practical application.
What we are dealing with here: The custom of reading three megillos on the three festivals, the question of why Shir Hashirim was specifically fixed for Shabbos, and the question of whether a berachah is recited over these readings.

The three megillos read on the festivals

The well known megillos: Everyone knows that we read Megillas Esther on Purim and Megillas Eichah on Tishah B'Av.
The three other megillos: On each of the three festivals there is a custom to read a megillah from Kesuvim: Shir Hashirim on Pesach, Koheles on Sukkos, and Rus on Shavuos.
When there is no Shabbos in Chol Hamoed: According to the Rema, when there is no Shabbos during Chol Hamoed, the megillah is said on the last day of Yom Tov. This year Shir Hashirim is said on Shevi'i shel Pesach, which falls on Shabbos.

The reasons for reading the three megillos

"It is customary to say Shir Hashirim, because the Exodus from Egypt is stated explicitly in it. And Koheles on Sukkos, because these are days of joy, and it is written in Koheles, 'and of joy, what does it accomplish?' And Rus on Shavuos, because the Yalkut on Rus states: what is the connection between Rus and Shavuos, that it is read at the time of the giving of the Torah? To teach you that the Torah is given only through suffering and poverty."
(Mishnah Berurah, s.k. 17)

The Exodus in Shir Hashirim: The reference is in the verse "To a mare among Pharaoh's chariots have I likened you, my beloved." That is only a single verse, and Shir Hashirim as a whole is expounded at great length, but the Mishnah Berurah takes the simple reason.

Sukkos as zman simchaseinu

Joy on all the festivals: "And you shall rejoice on your festival" is stated regarding all three festivals, and all of them are days of joy.
The added joy on Sukkos: On Sukkos joy is mentioned more than once, on Shavuos it is mentioned once, and on Pesach it is not stated explicitly at all. Nevertheless, the festivals are compared to one another in a number of areas, so there is joy on Pesach as well.
Simchas Beis Hasho'evah: The added joy on Sukkos also stems from the Simchas Beis Hasho'evah, about which it is said, "Whoever did not see the Simchas Beis Hasho'evah never saw joy in his life." That is why Sukkos is called "zman simchaseinu," the time of our rejoicing.
The agricultural dimension: The festivals in Eretz Yisrael are also agricultural festivals: Pesach is the beginning of the harvest, Shavuos is the time of the harvest, and Sukkos is the festival of the ingathering. By Sukkos the farmer has already gathered in his entire crop and sees that the year was a blessed one, and therefore this is the greatest time of joy.
"Chag" without a name: In the Gemara, when "chag" is mentioned without specifying which one, the reference is to Sukkos, from the word chagigah. Even though the chagigah offering is brought on all three festivals, the title "chag" became the exclusive designation of Sukkos.

Koheles: tempering the joy

The difficulty: Why do we read Koheles precisely at the time of the greatest joy?
The answer: Because it says in it, "and of joy, what does it accomplish?" There is a joy of mitzvah and one must rejoice, but one must not step beyond the bounds of what is permitted.

"Rejoice, young man, in your youth... but know that for all these God will bring you to judgment."
(Koheles)

Why the sefer was not suppressed: They sought to suppress Koheles and did not do so, precisely because of statements like this in it. And for that very same reason we read it on Sukkos, to remind a person of this point so that he not forget it: rejoice, but with measure.

(In passing: darshanim expound on the kesiv and the kri in this verse. The kesiv is "zeh" with a hei, which has the numerical value twelve, and the kri is "zu" with a vav, which has the numerical value thirteen. Some have found here an allusion to the celebration of a bar mitzvah at age thirteen and of a bas mitzvah at age twelve, two celebrations that have no explicit basis in the sources.)

Additional reasons for reading Koheles on Sukkos

The Hakhel gathering: Some say that Koheles was delivered by Shlomo Hamelech at the Hakhel gathering, and Hakhel takes place only on Sukkos, so Koheles fits Sukkos.
Seven and eight: Koheles says, "Give your portion to seven and even to eight," and only Sukkos has eight days.
The simplest explanation: Of all these reasons, the simplest is the tempering of joy: everything must be in measure.

Rus on Shavuos: the reasons

Suffering and poverty: As the Yalkut quoted by the Mishnah Berurah states, the Torah is given only through suffering and poverty, as we find with Rus the Moabite.
Conversion: Rus the Moabite converted, and the Jewish people too were like converts at the giving of the Torah.
Chesed: It is said that the Torah begins with chesed and ends with chesed, and so too Megillas Rus: it begins with chesed and ends with chesed.
The passing of Dovid Hamelech: Dovid Hamelech, who descended from Rus, passed away on Shavuos.
The time of the barley harvest: The episode of Rus and Boaz took place at the beginning of the barley harvest, which is the season of Shavuos.

Why Shir Hashirim is said specifically on Shabbos

The difficulty: If the reason is that the Exodus is stated explicitly in it, it should properly be said on the first day of Pesach, when they left Egypt, or on Shevi'i shel Pesach, when the sea was split. Fixing it for Shabbos does not fit that reason.

"On the Shabbos of Chol Hamoed: because the piyutim of that Shabbos are built upon Shir Hashirim, as above."
(Mishnah Berurah, s.k. 18)

What emerges from his words: This is not a reason for the custom but a proof of how ancient it is. The piyutim of the Kalir and the early paytanim that are said on the Shabbos of Pesach are built on the verses of Shir Hashirim, which shows that already then Shir Hashirim was said on Shabbos. The reason itself is not spelled out in his words.
The Be'er Heitev's opposite direction: The Be'er Heitev writes it the other way around, and the result is the same:

"On Shabbos we say in the Yotzer the piyutim of Shabbos Chol Hamoed, because Shir Hashirim is said on that day."
(Be'er Heitev, os 6)

A note from scholars of piyut: According to most contemporary scholars of piyut, these piyutim were originally composed for the first day of Pesach, and most of the paytanim lived in Eretz Yisrael, where there is no second day of Yom Tov. It was the community that later transferred them to Shabbos. If so, they are no proof that Shir Hashirim was said on Shabbos.
In our times: We do not say these piyutim, so the reason has no practical bearing. The main value of the point is as evidence for the antiquity of the custom.
The approach of the Pri Megadim: On the first day of Pesach there are many piyutim and the davening is long, and they did not want to lengthen it further. They therefore moved the saying of Shir Hashirim to Shabbos, to ease the burden on the congregation.
A further reason (the Gra and others): On Shabbos a large crowd comes to shul, even more than on Yom Tov, and therefore the reading was fixed for Shabbos.

Three customs regarding the saying of Shir Hashirim

On the Shabbos of Chol Hamoed: And when there is no Shabbos during Chol Hamoed, on Shevi'i shel Pesach that falls on Shabbos.
At the end of the Haggadah: Chassidim and men of action have the practice of saying it at the end of the Haggadah.
On Erev Shabbos: Chassidim and men of action have the practice of saying it before Kabbalas Shabbos, and there are various customs here: before Minchah, after it, or between Minchah and Ma'ariv.
Other occasions: There are those who say it on additional occasions as well.

Which custom is the earliest

The early custom: Saying it on Shabbos, which already appears in Maseches Sofrim, a work no later than the era of the Geonim. There it says explicitly that it is said on Shabbos, and according to some versions even on Motzaei Shabbos, and it does not specify the Shabbos of Chol Hamoed in particular.
The other two customs: These are later, and their source is apparently in Kabbalah. It is not clear which of them came first, and it seems that saying it on Friday night preceded saying it at the end of the Haggadah, but there is no decisive evidence.

The berachah over the reading of the megillos

The two texts of the berachah: The text "al mikra megillah" is familiar from Megillas Esther, and seemingly that is the text appropriate for any megillah. Where, then, does the text "al mikra kesuvim" mentioned by the Rema come from? From the fact that this berachah is brought in Maseches Sofrim.
A proof of antiquity: The very fact that the Rema mentions the berachah "al mikra kesuvim" reinforces what was said above, that saying the megillah on Shabbos is the earliest of the three customs. Since the berachah is brought in Maseches Sofrim, the Rema comes and says that we do not say "al mikra kesuvim," and in his view we do not even recite "al mikra megillah."
The Mishnah Berurah's position: He agrees with the Rema that no berachah is recited.

(The expression "the people have adopted the practice" itself deserves study: some have seen in it language indicating a custom with which the posek is not entirely comfortable. This requires checking whether he uses this expression elsewhere, and one should not decide the matter by conjecture.)

"And the Magen Avraham agreed with those poskim who require a berachah over all of them except for Koheles, and the Gra in his commentary wrote that one should recite a berachah even over Koheles, as is written in Maseches Sofrim."
(Mishnah Berurah, s.k. 19)

Why the Magen Avraham excludes Koheles: The Gemara itself discusses which books they sought to suppress, and regarding Koheles they certainly sought to suppress it and did not. Since there is an element of difficulty with it, no berachah is recited over it.
The Gra's approach: A berachah is recited even over Koheles, as is written in Maseches Sofrim. And not only that: in his view one also recites Shehecheyanu, not only "al mikra megillah." There are those who follow the Gra only regarding the berachah "al mikra megillah" and do not say Shehecheyanu, but the Gra himself requires Shehecheyanu as well.
The practice in fact: The Luach Eretz Yisrael brings that many follow the Gra, and that is the practice in this shul: we recite "al mikra megillah" and say Shehecheyanu.
When the megillah is written on parchment: The Mishnah Berurah writes that one who makes a berachah when the megillos are written on parchment should not be stopped, since there are opinions that once the megillah is on parchment it is entirely in order. Some have even suggested that the Rema himself might concede this, and that his objection to the berachah was not from the essence of the law but because in his time the megillos were not written on parchment.
The Gra's position on this: The Gra recites a berachah even when the megillah is not written on parchment, for if we recite berachos over the haftaros, why should we not recite them over these? The community did not accept this, as the Mishnah Berurah says, but when the megillah is on parchment one may certainly recite the berachah.

Background: the berachah over haftaros not written on parchment

Haftaros in earlier times: The haftaros too were written on parchment, and because of the community's financial difficulty their writing shifted to print.
The dispute: There was a major dispute about whether a berachah may be recited over a printed haftarah, since over a printed sefer Torah no berachah is recited. Some said one may not recite a berachah unless it is written on parchment, and in practice the halachah was decided that one may recite the berachah.
Shuls where the haftaros are on parchment: There are shuls where all the haftaros are written on parchment, such as the Great Synagogue, and also a shul in Tel Aviv. In such a place, since the haftarah is without vowels and cantillation marks, no one would be able to come up and read it other than the ba'al korei.
On this basis the Rema's position is understandable: With the haftaros they were lenient, but with the reading of the megillos, which is not a simple reading, he was not lenient.

(Raising such a proposal in practice depends on finding a donor, and in any case it stands to reason that even if it were written on parchment people here would come up and read, since they know how to read by heart and without cantillation marks.)

Saying Shir Hashirim at the end of the Haggadah and on Erev Shabbos

At the end of the Haggadah: This is readily understood: if Shir Hashirim alludes to the Exodus from Egypt, its place is at the Seder.
The connection to Shevi'i shel Pesach: Since Shir Hashirim is also said on Shevi'i shel Pesach, when the final departure from Egypt took place (whereas on Seder night Pharaoh was still pursuing them), there is a special reason to say it at the end of the Haggadah as well.
On Erev Shabbos: Here the reason is more complex. One of the simpler explanations: according to Rabbi Akiva all the songs are holy and Shir Hashirim is the holy of holies, the greatest elevation of all. Shabbos too is the greatest sanctity of all times, and therefore we say Shir Hashirim at the transition from a weekday to the holiest day.
Shabbos and Yom Kippur: Yom Kippur is called "the holy day" because it comes once a year and the feeling on it is more intense. But in terms of the sanctity itself, the punishment for desecrating Shabbos is greater than the punishment for desecrating Yom Kippur, which shows that the sanctity of Shabbos is the greatest of all.

In passing · the BaHaB fasts after the festivals (Siman 492)

"Some have the practice of fasting on Monday, Thursday and Monday after Pesach, and likewise after Sukkos."
(Shulchan Aruch, Siman 492)

"Because there is a concern that in the course of feasting and rejoicing they may have come to sin, as we find with Iyov, who brought offerings, and a fast takes the place of an offering. But regarding Shavuos, which is only one day, there is no such concern."
(Mishnah Berurah, s.k. 1)

The practice of those who are stringent: Some have the practice of fasting after Shavuos as well.
A proof that Pesach is a time of joy: The very custom of fasting after Pesach teaches that Pesach too is a time of joy, and therefore there was a concern that in the midst of the joy people stumbled in something, and the fast comes to atone for it.
The conceptual conclusion: From here we see that there is a concern about excessive joy that goes beyond the permitted bounds. Therefore in Koheles a person is reminded to rejoice, but with measure and within the permitted bounds.