Why does Moshe's name not appear in the entire Parshah, as it does in every other Parshah in the Torah from the time that he was born?
Ba'al ha'Turim #2: Because Tetzaveh deals with topics that concern the Kehunah, which saddened Moshe, 3 since the Kehunah Gedolah was originally intended to come from him, a Z'chus that he lost during the episode of the 'burning bush', when he refused to become Hashem's emissary.
The reason that Hashem erased his name specifically from this Parshah is because it always corresponds with Moshe's Yohrtzeit (commentaries).
And it is to demonstrate Moshe's sadness that the Torah omits his name from the entire Parshah.
Why did Hashem say "Tetzaveh" in the future"
Rashi (in Vayikra 24:2): Because the Mitzvah of kindling the Menorah is contained in the opening Pesukim in Emor. The Pasuk here is merely presenting details concerning the Menorah that was discussed in Terumah.
Seforno: (In a similar comment on the Lashon "Veyikchu Eilecha" [in the future]): Hashem is instructing Moshe here that, when the oil that they donated in Terumah runs out, they should donate fresh oil to continue with the Mitzvah. 1
And not assume that the Mitzvah is only temporary, and will terminate when the original batch of oil runss out (Seforno). See also Ba'al ha'Turim as to why the Torah uses the Lashon Tzav both by the Menorah and by the Korban Tamid in Vayikra (6:1). Similarly, the Rashbam points out that wherever the Torah the expression "Tzav", it means that the Mitzvah is to last for all generations - in this instance, it entails donating fresh oil every year.
And why did He use the expression "ve'Atah Tetzaveh" instead of "ve'Asisa" that He used constantly in Terumah?
Ramban #1: Because there was no way that the people could produce in the desert oil that was 'Zach'; It had to have been prepared beforehand and was now brought to Moshe 1 to check (for Zach and Kasis). 2
Ramban #2 and Seforno: Because, whereas "ve'Asisa" implies via somebody else, Hashem said "ve'Atah here, because he wanted Moshe to personally issue Yisrael with the instruction on manufacturing the oil, 3 bearing in mind that the actual production of the Begadim was performed by Betzalel. 4
Perhaps it is because Moshe's name is not mentioned in the Parshah 5 that the Torah stresses a number of times "ve'Atah", to emphasize that ultimately, everything that took place in the Mishkan was under his authority.
In fact, the Ramban adds, it was the princes who brought it.
Which explains why the Torah writes "Eilecha" (Ramban).
And it is for the same reason that Hashem shortly uses the expression "ve'Atah Tetzaveh" with regard to bringing Aharon and his sons close (Shemos 28:1) to personally inaugurate them, and with regard to instructing the wise-hearted men the details of manufacturing the Bigdei Kehunah, (Shemos 28:3 [Ramban and Seforno]) - because he knew the exact task that each person was capable of performing (Ramban [Refer also to 28:3:1:2 and refer to 28:3:1:3]).
How does one obtain good quality oil that falls under the sategory of "Zach"?
Rashi #1: 1. By picking the olives from the very top of the tree. 1
Rashi #2 and Rashbam: By pounding rather than grinding them (Kasis), and using only the first drop of oil that emerges (Rashi). 2
What are the implications of "Kasis la'Ma'or"?
Rashi (in Menachos, 86b): It implies that the oil for the Menachos may be ground. 1
Though if the olives are beaten, the oil is not Pasul (See Gemara there).
Why does the Torah use the word "Leha'alos" rather than 'Lehadlik'?
Rashi: Because the Kohen who lit the Menorah was obligated to apply the lighter until the flame was rising by itself.
What are the connotations of "Tamid" with regard to the Menorah?
Rashi: It means every night. 1
In similar vein to (the Olas Tamid [Rashi in Chagigah, 26b], to) the Minchas Chavitin (of the Kohen Gadol [in Vayikra 6:13]), where it means half each morning and half each afternoon; as opposed to the "Tamid" that is written by the Lechem ha'Panim, where it means that they had to be on the Shulchan from Shabbos to Shabbos (Rashi).
See Ramban, who elaborates on the definition of 'Ner ha'Ma'aravi'.