Why is B'ris Milah performed on the eighth day? Perhaps he will die without Milah!
Hadar Zekenim #1, Moshav Zekenim (from Nidah31b), Targum Yonasan: It is so his mother can immerse and become permitted to his father, and all are equally happy when they do the Mitzvah. 1
Hadar Zekenim #2: A parable for this is the king's friend who came to see him. The king ordered to prepare a feast for him, but said to wait until a beautiful matron comes. Also here, Hashem wants the baby to see Shabbos first. 2 Similarly, a Korban is acceptable only from day eight for this reason.
Seforno: Because this is when the Dam Nidus from which the fetus was fed has been digested, and the baby is Tahor and can enter Bris Kodesh.
Peirush Yonason: This is why it was taught here amidst laws of Tum'as Yoledes (even though it was taught above - EC). The Torah fixed the Bris for the eighth day even when she cannot become Tehorah, e.g. Yoledes b'Zov, or a twin girl was born with the boy
Why is B'ris Milah performed only on the eighth day?
Seforno: Because prior to that the Dam Nidus has not yet dried up, in which case the baby is still adversely affected by it.
Targum Yonasan: Refer to 12:3:1:1*.
If Hashem wants that we be circumcised, why is a boy not born circumcised?
Moshav Zekenim: R. Akiva answered Turnusruus, Hashem gave to us Mitzvos to purify us and receive reward.
In Shabbos (132b), we expound that Bris Milah overrides Shabbos. A Kal v'Chomer already teaches this! Milah overrides Avodah, and Avodah overrides Shabbos!
Moshav Zekenim: The Kal v'Chomer can be challenged. Only Korbanos Tzibur override Shabbos. Milah is [like] a Korban Yachid.
Why do we need "uva'Yom [ha'Shemini]"to teach that Milah overrides Shabbos (Shabbos 132b)? Even if it were not a Mitzvah, it is mere cutting flesh, like it says about a baby born Bein ha'Shemashos! (136a)
Moshav Zekenim, citing 136a: It is not needed to permit Milah of a baby born Bein ha'Shemashos; in any case it is permitted. 1 It is needed according to R. Yehudah, who permits Melachah for Machshirei (things needed to enable) Milah. They are permitted only for a Vadai Mitzvah.
Moshav Zekenim citing R. Simchah: The verse excludes an Androginus and someone born circumcised. Such Safek Milah does not override Shabbos. Ba'Yom excludes the [eighth] night. The prefix Vov teaches that even after eight days [Milah must be during the day]. However, why don't we know this already from "u'Ven Shemonas Yamm Yimol" (Bereishis 17:12)?
Moshav Zekenim: The verse teaches that the entire day is Kosher for Milah, just zealous people do it promptly. However, Pesachim 4a teaches [that one must] to do it in the morning! 2
The Gemara said so about a Safek Nefel! If it is Chai, fine (he properly circumcises). If not (rather, it is a Nefel), he merely cuts flesh. How did Moshav Zekenim understand this to discuss a baby born Bein ha'Shemashos?! Perhaps in his text, the 'Yud' in 'Chai' looked like ' (an apostrophe). His text read 'if it is Ches' (the eighth day), fine. If not, he merely cuts flesh.' This is difficult if birth was when Shabbos entered; if next Shabbos is really day nine, would we call Milah 'mere cutting flesh'?! Rather, birth was when Shabbos ended; if next Shabbos is day seven, there is no Mitzvah, and he merely cuts flesh. Why is this called 'mere cutting'? Due to this, he loses the Mitzvah of Milah tomorrow! Perhaps he holds that Hatafas Dam Bris is as great as Mitzvas Milah. Or, indeed, he loses the Mitzvah; regarding laws of Shabbos, it is mere cutting flesh. (PF)
It is not clear what his question is. Ba'Yom teaches that the Mitzvah applies the entire day, just Zerizim do it in the morning! (PF)
Why does it say "Yimol Besar Orlaso"?
Moshav Zekenim: It says about Tzara'as "b'Or Besaro". We learn that even if there is Tzara'as on the foreskin, he cuts it. Normally, a Lav forbids cutting off Tzara'as.
What is the reason for Bris Milah?
Moshav Zekenim: citing Rambam (Moreh ha'Nevuchim 3:49): It is to curb excess lust and diminish the amount that one will engage in Bi'ah.
Moshav Zekenim: It is a sign that one believes that Hashem is one. One might want to pretend that he believes, in order to get some benefit, but he would not circumcise himself or his son for this. There is great love and mutual help among people with the same sign.