What is the significance of the seven days of Sukos followed by the eighth day?
Moshav Zekenim: Sukos is for seven days, for there were seven Clouds of Glory - one above, one below, four in the four directions, and one that went in front of and led Yisrael in the Midbar.
What is the reason for the repetition "v'Chagosem Oso Chag"?
Moshav Zekenim: Yaravam invented a Chag in Cheshvan. The Torah warns to celebrate the Chag only in the proper time of the Chag.
Why does it say "v'Chagosem Oso Chag la'Shem"?
Moshav Zekenim: We could expound that the [Korban] Chagigah is to Hashem, but the Sukah is not - however, it says in Beitzah (30b) that Shem Shamayim takes effect on the Sukah. The wood is forbidden all seven days due to Muktzah for the Mitzvah, or [to avoid] disgrace of the Mitzvah.