Seeing as "u've'Mo'adeichem" refers to the Yamim-Tovim, what does "u've'Yom Simchaschem" refer to?
Sifri #1: It refers to Shabbos. 1
Sifri #2 (citing R. Nasan): It refers to the Korban Tamid. 2
Ibn Ezra: Both of these refer to days fixed for Simchah - after returning from the enemy's land, or defeating an enemy that came upon you - Purim and Chanukah.
Why is the word "u?ve?Rashei Chodsheichem" missing a 'Yud'?
Sukah, 55a: To teach us that the months of the year are all the same - that if Rosh Chodesh falls on Shabbos one blows only one set of Teki'os at Musaf, like every other Rosh Chodesh, and not two.
What are the connotations of "Al Oloseichem"?
Rashi: It means that the Kohanim blow the trumpets when the Olos Tzibur are brought on the Mizbe'ach. 1
See Sifsei Chachamim.
What is "Zivchei Shalmeichem" referring to?
It refers to the two lambs that the Tzibur 1 brings on Shavu'os as a Shelamim together with the Sh'tei ha'Lechem.
See Torah Temimah, note 10.
What is the significance of the comparison between "Zivchei Shalmeichem" and "Oloseichem"?
Zevachim, 55a: It teaches us that the Zivchei Shalmei Tzibur are Shechted in the north, like Olos. 1
Arachin, 11b: It teaches us, on the one hand that the Zivchei Shalmei Tzibur are Kodshei Kodshim, and on the othe that, like the Zivchei Shalmei Tzibur, the Olah has a fixed time - to preclude the Olos Nedavah shel Tzibur, which do not. 2
How will the blowing the trumpets result in being remembered for the good before Hashem?
Targum Yonasan: By causing the Satan to become confused and unable to prosecute. 1
Whenever the trumpets are blown - even on days other than Yom Kipur (when in any case he cannot prosecute).
Why did the Torah switch from "Ve'nizkartem Lifnei Elokeichem in Pasuk in Pasuk 9, to "Vahayah lachem le'Zikaron" here?
Oznayim la'Torah: Because whereas Pasuk 9 is referring to Hashem remembering Yisrael and saving them from their eneimies, the current Pasuk is referring to Yisrael remembering Hashem - by blowing the Shofar on Rosh Hashanah. 1
Which is hinted in the word "u've'Roshei Chodsheichem", which is written minus a 'Yud'. See Oznayim la'Torah, who elaborates at length.
Why does the Pasuk conclude "Ani Hashem Elokeichem"?