Seeing as Hashem issued the instructions to construct the Mishkan in Tishri, why did Moshe only set it up in Nisan?
Tanchuma: Because Nisan is the month in which Yitzchak (the symbol of Avodah) was born.
Oznayim la'Torah (based on Tosfos, Rosh Hashanah, 27a): Because the Mishkan is compared to the creation of Heaven and earth (Tanchuma), and Hashem planned to create the world in Tishri (like R. Eliezer) but He only created it in Nisan (like R. Yehoshua).
Why does the Torah write "Vay'hi ba'Chodesh ha'Rishon ... be'Echad la'Chodesh" and not simply "Vay'hi ba'Shanah ha'Sheinis be'Echad la'Chodesh ha'Rishon ... "?
Oznayim la'Torah (based on Shabbos, 87b): It hints at the importance of the day, which took ten crowns (firsts). 1
Refer to Vayikra, 9:1:3:1*.
Why did the Torah not give detailed instructions as to how to erect the Mishkan (in the way that Moshe subsequently erected it - starting from Pasuk 18)?
Ramban: Because Hashem had already shown him exactly how it was to be done. 1 And it is by the same token that the Torah is brief concerning the instructions regarding the Shulchan and the Menorah 2 and the Begadim of Aharon and his sons in Pesukim 13 and 14. 3
Perhaps it is because the Torah writes "es Mishkan Ohel Mo'ed" - implying the Heichal, which contained the Shulchan, the Menorah and the Mizbe'ach ha'Ketores, and the Torah says nothing about the Avodah concerning these three Keilim untim the eighth day. 4
Why does the Torah write that Moshe should set up the Mishkan on Rosh Chodesh Nisan (the eighth day of the Milu'im), when in fact, he first set it up on the first day of the Milu'im, seven days earlier?
Ramban and Moshav Zekenim: What the Pasuk means is that on Rosh Chodesh Nisan, Moshe will erect the Mishkan for the last time. 1 It will stay erect until they travel, and then the Levi'im will dismantle it and set it up.
In fact, Hashem had instructed him to set it up (See 26:30), as soon as it was completed, and he extrapolated from this Pasuk that when he would set it up prior to Rosh Chodesh Nisan, it would not be on a permanent basis, but that he would have to take it down again each morning (some opinions say twice or even three times a day). See Ramban DH 'be'Yom ha'Chodesh' and 'u'Kevar Darshu'.
Why was Moshe required to dismantle the Mishkan each morning (Ramban) before putting it up, each day during the seven days of Milu'im?
Ramban: Either to teach the Levi'im how it had to be done, or in order to crown Rosh Chodesh with the crown of the inauguration. 1
See opening Rashi in Shemini. See also Ramban for a number of reasons as to why the Mishkan had to remain intact throughout the day and the following night, and could only be dismantled first thing in the morning.