7 represents the natural world, the world was created in 7 days, 7 musical notes, 7 colors in the rainbow. Shabbos is the seventh day of the week.
Sources 1-4 are from the reading for Sukkoht from the Torah. The holidays are presented in chronological order from creation.Shabbos (7 Days) starting from creation. Then the creation of the Jews during Yitziyat Mitzrayim (Paysach) to Matan Torah (Shavuot) is seven weeks. Then Seven months from Yitziyat Mitzrayim is Sukkoht. This lasts seven days.
God desires an abode in this world, this abode is the Jewish people. God’s presence is manifest in the Jewish people's life through holidays. The Sukkah is a physical representation of Gods presence,protection and providence in this world made possible through physical means that is why it is 7 seven days.
God wishes to be known through nature, through the number seven. Through the name Elokim which is equivalent in Gematria to Teva or nature. We read Kohelet on Sukkoht, the Safer calls Hashem Elokim.
The Zohar teaches, only through the natural order are miracles manifest. This is the secret behind the seven years Yackov had to work for Leah then Rachel. In order to understand the Drasha you must go through the Pshat.
I will use one aspect from each of the Haftorot for Sukkot.
Earthquakes: The physical manifestation of God through nature. (Source 5 )
The Beis Hamikdash: God’s abode in this world (Source 6 )