When did they perform Nisuch ha?Mayim?
Ramban and R. Bachye: They performed it on each of the seven days of Sukos (in the morning) - Halachah le?Moshe mi?Sinai. 1
See R. Bachye for details concerning Nisuch ha?Mayim and its performanced.
What is the reason for Nisuch ha?Mayim?
R. Bachye (citing Rosh Hashanah 16a): In order to receive Hashem?s blessing for rain. 1
Seeing as the rain-season begins after Sukos.
Why, on some days, is the Chatas called "Se'ir", and on others, "Se'ir Izim"?
Hadar Zekenim (on Pasuk 13): The Siman is 'ABaD Izim' - The Torah writes 'Se'ir Izim' on days 'Alef,' 'Beis' and 'Dalet'. Yisrael are Azim (most brazen) of the nations. These days are the order in which our Avos was born - Avraham was first, Yitzchak second, and Ya'akov fourth. Eisav was third (and he is called Se'ir)!
R. Bachye (Vayikra 1:5): Se'ir Stam is a yearling. Se'ir Izim is above one year.
Rosh (on 28:9): It says "Se'ir Izim" on days one, two and four. The total number of bulls offered on those days is thirty-five, the same as the days that it says "Se'ir".
Kol Eliyahu and Divrei Eliyahu citing the Zohar: Refer to answer #3. Thirty-five nations nurture from Yishmael, who is called 'Se'ir Izim', and thirty-five from Eisav, who is called 'Se'ir'..
