What is the exact meaning of "P'ri Eitz Hadar"?
Moshav Zekenim (from Yerushalmi Sukah 3:5): Its wood resembles its fruit (Korban ha'Edah - in beauty).
What are the implications of the word "Hadar"?
Ramban (citing Ibn Ezra) and Targum Yonasan: It means that there is no tree fruit more beautiful than the Esrog.
Ramban, Moshav Zekenim and Targum Onkelos: 'Esrog' is the Arama'ic translation of "Hadar". 3
According to the Ramban, this D'rashah and the previous one are Asmachtos to support the Chachamim's traditional interpretation of the Pasuk.
See Sifsei Chachamim.
What do we learn from "Kapos"?
Rashi: It is written Chaser (without a 'Vav') to teach that only one Lulav needs to be taken. 1
Moshav Zekenim: If its leaves are spread, Yekaftenu (bind it).
See Sifsei Chachamim.
What is the exact meaning of "Anaf Eitz Avos"?
Rashi: It means that its branches are plaited like ropes - with reference to a Hadas, whose leaves are formed in groups of three like plaits.
How many of each species must one take?
Ramban (like R. Akiva, Sukah 34b): One of each species - "One fruit from an Esrog-tree, one Lulav-branch, one myrtle-branch and one branch from a willow-tree that grows by the brook". 1
R. Yishmael (Sukah 34b): One Esrog, one Lulav, three myrtle-branches and two willow branches.
Ramban: And the reason that Onkelos translates them all in the plural is because it goes with "u'Lkachtem Lachem", which is plural.
What is the underlying reason behind the Mitzvah of the Arba Minim?
Da'as Zekenim #1, Hadar Zekenim: It is a sign that we were victorious in the judgment of Yom Kipur against the other nations. If one saw two parties come in front of the king for judgment, and one leaves with a white stick or apple, it is known that he won.
Da'as Zekenim #2: Of the four Minim, the Esrog has taste and smell, corresponding to a Tzadik with Torah and good deeds. The Lulav comes from a tree with [dates, which have] taste but not smell, corresponding to a Beinoni with good deeds but not Torah. Myrtle has smell, but not taste, corresponding to one with Torah, but not good deeds. Willow has no taste or smell, corresponding to an ignoramus which is not good deeds or Torah. We tie the four 5 together, to hint that Hashem is not appeased with Yisrael until all are in one assemblage. "Ha'Boneh va'Shamayim Ma'alosav va'Agudaso Al Eretz Yesadah" (Amos 9:6) - Hashem is elevated when all become one assemblage.
Da'as Zekenim #3: The Lulav is like a spine, myrtle [leaves] resemble eyes, willow [leaves] resemble lips, and an Esrog resembles the heart. There is no limb greater than these. David said "Kol Atzmosai Tomarnah Hashem Mi Chamocha" - one must praise Hashem with the mouth, heart, eye and spine, for the entire body depends on the spine.
Moshav Zekenim: The Etz ha'Da'as was an Esrog. An Esrog should atone for a sin through an Esrog.
Vayikra Rabah (30:9, partially cited in Moshav Zekenim): All of them refer to Hashem. Pri Eitz Hadar refers to Him - "Hod v'Hadar Lavashta." Kapos Temarim - "Tzadik ka'Tamar Yifrach." Anaf Eitz Avos - "v'Hu Omed Bein ha'Hadasim." Arvei Nachal - "Solu la'Rochev ba'Aravos b'Kah Shmo." 6
Which we do with four species that require a lot of water (Ta'anis 2b).
Ramban: With which it is not tied because it is only with it that Adam and Chavah sinned. And by the same token, Shemini Atzeres, which represents the Esrog, is a 'Regel bi'Fnei Atzmo' (R. Chavel's footnotes DH 'Im She'ar ha'Minim'). And the reason that we do not take the Arba Minim on Shemini Atzeres is because Shemini Atzeres itself represents the Esrog.
See Ramban DH 've'Ta'am ha'Mitzvah'.
In Menachos (27a), R. Yehudah says that one is Yotzei only if all four are bound together. Teshuvas Avnei Nezer (OC 491) - R. Yehudah holds that one holds all four in one hand, but he agrees that the Esrog is not tied to the other three. Moshav Zekenim - we do not bind the Esrog with the others, for there is no Vov before "Kapos" (to connect them). We bless on the Lulav, and take it in the right hand, for [two] other species are tied with it. Refer to 23:40:151:4.
Michtav me'Eliyahu (5, p.418): These are perceptions of Hashem in the four worlds; each species corresponds to one letter of His name. The Mitzvah is to unite them, i.e. to put them to your heart.
What do we learn from "u'Lkachtem Lachem?
Hadar Zekenim: Take them to stroll with them. 3
Sukah 34b: We require Lekichah Tamah (pure, complete) - every species is Me'akev (one fulfills the Mitzvah only if he takes all four). Moshav Zekenim - Lekichah Tamah requires taking them without an interruption between his hand and the Minim.
Moshav Zekenim #1: We learn from a Gezeirah Shavah from "u'Lkachtem Lachem" written about the hyssop bundle [used to sprinkle Dam Pesach in Mitzrayim - Shemos 12:22]. Just like there they were tied together, also here. The Halachah follows Chachamim, who do not require binding the Lulav, only it is a Mitzvah to beautify it - "Zeh Keli v'Anvehu."
Moshav Zekenim #2: It atones for you.
Moshav Zekenim: Partners are not Yotzei. If brothers bought an Esrog from the estate, they are Yotzei only if one could eat it and the others would not be upset (Bava Basra 137b). The Rashbam said that now that the Tzibur buys one Esrog, this is unlike brothers. They bought with intent [that each one own it when he takes it]; it is like a gift on condition to return it. The case of brothers is when they bought it to eat it.
Da'as Zekenim gives a nice parable to explain why theft is Pasul for Mitzvos.
Sukah 41b: The custom in Yerushalayim was to hold one's Lulav while going to the Beis ha'Keneses, during Keri'as Shma and Tefilah, while visiting the sick and consoling mourners
Why does it say "Yom ha'Rishon"? It is not the first of the month!
Hadar Zekenim citing Tanchuma, Moshav Zekenim: It is the first day in the calculation of Aveiros, i.e. the first Regel after Yom Kipur. 1 Tur (OC 581) - after Yom Kipur people are busy with the Mitzvos of Sukah and Lulav, and do not do Aveiros.
Rosh: Hashem forgives everything on Yom Kipur. Yisrael take their Lulavim and praise Him on Sukos, and He is appeased and pardons them, and now is the the first day in the calculation of Aveiros. (Perhaps now he pardons Aveiros between Yom Kipur and Sukos. - PF)