What are the ramifications of "Tachogu"?
Rashi: It refers to the obligation to bring a Korban Shalmei Chagigah on Yom-Tov.
Ramban, Moshav Zekenim: It implies giving thanks to Hashem on the successful ingathering of the crops.
What does the word "Ach" come to preclude?
Rashi: It comes to preclude Shabbos from the Shalmei Chagigah. 1
Rashbam: It comes to tress the distinction between Rosh Hashanah and Yom Kipur, which come as a remembrance and as an atonement, and Sukos, that comes to rejoice on it and to thank Hashem for filling one's house with all good things during the ingathering of one's crops.
Seforno: After mentioning the issues that apply equally to all the Yamim-Tovim - such as that they are all Mikraei Kodesh and that they require a Korban Musaf, the Torah now presents the innovations that pertain to Sukos exclusively - the eighth day of Yom-Tov, 2 the Sukah and the four species.
Hadar Zekenim: It excludes Pesach and Shavu'os. Then, one need not stay in Yerushalayim more than one day 3 , for one is distracted with the need to harvest his crops. However, on Sukos, one has already gathered the Peros into his house, and Yom Kipur passed and your sins were pardoned, you should increase Simchah on Simchah and [stay to] engage in it.
Seforno: Unlike the days of the week, the weeks and the years, all of which revolve around the number seven.
Rosh Hashanah 5a says that Chag ha'Matzos requires Linah, for it says "u'Fanisa ba'Boker v'Halachta l'Ohalecha." The Ritva and some others explain, unlike Hadar Zekenim, that this refers to Linah all seven days. (PF)
What is the significance of the phrase "b'Aspechem es Tevu'as ha'Aretz"? Why does the Torah insert it?
Rashi (from Toras Kohanim): It teaches us that Sukos must coincide with the ingathering 1 of (most of) 2 the produce from the fields (in the autumn). Consequently, Beis-Din must sometimes fix a leap-year - to avoid Sukos falling in the middle of summer or winter. 3
If "Tachogu oso" implies only one Korban, what are the implications of "Shiv'as Yamim"?
Rashi and Ramban (citing the Toras Kohanim): It teaches the law of Tashlumin, that someone who fails to bring the Chagigah on the first day of Yom-Tov, brings it on one of the subsequent days.
Why is Simchah mentioned three times regarding Sukos, once regarding Shavu'os, and not at all regarding Pesach?
Da'as Zekenim: On (at the time of) Pesach, one has not harvested any grain. On Shavu'os, one has gathered the grain, but not tree fruits. On Sukos, all was gathered into his house, and his sins were pardoned [on Yom Kipur], it says three Simchos - for grain, fruits and pardon of sins.