The Gemarah Sukkah (11b) says תניא לולב מצוה לאוגדו ואם לא אגדו כשר אי ר' יהודה כי לא אגדו אמאי כשר אי רבנן אמאי מצוה לעולם רבנן היא ומשום שנאמר (שמות טו, ב) זה אלי ואנוהו התנאה לפניו במצות. Usually, הדור מצוה is something added to the mitzvah, why then does the beratah indicate there is something lacking in the mitzvah if the lulav is not tied to the other species? Rashi Sukkah (33b) is of the opinion that the Rabbanan also hold that one needs a real tie of the four species. Why does one need a real tie if the point is just hiddur mitzvah? The Gemarah explains the machlokes is that R' Yehuda learns a גז"ש from אגודת אזוב that one needs a tie and the Rabbanan don't accept the gezerah shava. Even the Rabbanan that don't accept the gezerah shava hold the yesod of the hiddur is in the maaseh לקיחה for the species one is taking to be tied in a bundle. It is not a hiddur in the cheftah of the lulav but rather in the לקיחה of the lulav. That is why Rashi holds it needs to be a real tie in order to define it as a אגודה. This may be why the Rambam never cites the general law of hiddur mitzvah but he does cite the law of tying the lulav (Lulav 7:6) because this is not the regular law of adorning the cheftzah of a mitzvah but adorning the לקיחה.
What is the idea of binding everything together? The Midrash says (Emor 30:9) the 4 species all represent Hashem. We demonstrate through the binding of the species that we are totally bound to Hashem.
Why is it only after the holiday of Sukkot does Hashem say קשה עלי פרידתכם and we add a day to the holiday? And how does it help to add a day, that will just make the departure even harder? Why do we have 7 hakafot on Simchat Torah?
The Gemarah Taanit (7a) says אמר רב יהודה: גדול יום הגשמים כיום שניתנה בו תורה, שנאמר: "יערוף כמטר לקחי", ואין לקח אלא תורה, שנאמר: "כי לקח טוב נתתי לכם תורתי אל תעזובו". רבא אמר: יותר מיום שניתנה בו תורה, שנאמר: "יערוף כמטר לקחי", מי נתלה במי? – הוי אומר: קטן נתלה בגדול. What is the comparison between rain and Torah? The Kozhnitzer Maggid says the Gemarah is referring to Shemini Atzeret, the day we are judged for rain, which is as great or greater than Mattan Torah. Again, what is the comparison?
The holiday of Sukkot is the one holiday where we include the gentiles. We offer sacrifices for them as well. [The reason being that as opposed to the other festivals which commemorate our leaving Egypt and receiving the Torah which is unique to us, Sukkot also is an agricultural holiday celebrating the harvest and the celebration of nature is something the gentiles can appreciate as well - Meshech Chachma Pinchas.] Therefore, it needs to be emphasized at the end of the holiday that only we innately matter to Hashem. How does it help to add a day? The Shem MiShmuel explains that there are two types of kinyanim. There is a normal transaction between buyer and seller where the transaction happens and the two parties then walk away. That is the meeting of the gentiles with Hashem on Sukkot. They meet and move on. However, there is another type of kinyan. That is the kinyan of marriage. In such a type of kinyan, the parties do not turn around and walk away from each other, on the contrary they are now bound together wherever they go. That is our relationship with Hashem. Yes, eventually the music stops, but we take the energy of the Yom Tov with wherever we go. We designate one day to be alone with Hashme after the guests leave and we take that closeness with us. That may be the idea of the hakafor on Simchat Torah. Just as a bride encircles her husband seven times, so too we become united with the Torah and Hashem in the circle 7 times.
The Gemarah Taanit (10a) says א"י משקה אותה הקב"ה בעצמו וכל העולם כולו ע"י שליח שנאמר (איוב ה, י) הנותן מטר על פני ארץ ושולח מים על פני חוצות. The rain of E.Y. is called מטר, for other lands it's called מים. The word מטר is like the word מטרה, there is a target, a purpose. The rain that comes to E.Y. is a chesed from Hashem. The rain comes for its own purpose. The other lands get מים, they get water because they need to exist, it is not a chesed from Hashem. That is why the coming of rain or Shemini Atzeret is compared to Mattan Torah. Just as by Mattan Torah, Klal Yisrael was singled out to receive the Torah, the rain is targeted specifically for the needs of people. S.A. is the time when we are singled out above everyone else in the world to have a special closeness with Hashem and therefore it is likened to Mattan Torah (see Zeav Yitrof Sukkot simanim 70-71.)