Why does the Torah repeat "be'Etzem ha'Yom ha'Zeh" at the beginning of this Pasuk, having already mentioned it in Pasuk 17:23?
Rashi: To teach us that Avraham was circumcised at the end of his ninety-ninth year (i.e. at his ninry-ninth birthday), and Yishmael at the end of his thirteenth year. 1
Ramban: It teaches us that Avraham circumcised all three hundred and eighteen members of his household on the same day that he circumcised himself and Yishmael
Moshav Zekenim asks: Midrash Shir ha'Shirim learns from "k'Chom ha'Yom" (Bereishis 18:1) that Avraham's Milah was in Tamuz! (However, this does not appear in our versions of the Midrash. How would the Midrash explain "la'Mo'ed... ka'Es Chayah" (18:14)? Yitzchak was born on Pesach, which would neither be the next Mo'ed, nor the same time next year!
Whom did Avraham circumcise first
Ramban: He circumcised everyone else first (or personally oversaw other Mohalim that did so), and then circumcised himself last. After his Milah he would be in pain or sick, and unable to carry out the Mitzvah. This is implied by 17: 23-24, where it mentions Yishma'el's Milah and those of Avraham's household before his own. 1
Ohr ha'Chayim (to 17:23): Avraham circumcised himself first. The verse "Himol Yimol" (17:13) teaches that only one who himself is circumcised may circumcise others. 2
Rosh (to 17:13) Avraham tried to circumcise several members of his household first, and failed to do so safely. Mamrei gave Avraham the advice to circumcise himself first, after which he was able to perform the Milah on others. 3
Da'as Zekenim, Pirkei d'Rebbi Eliezer (29): Avraham summoned Shem ben Noach to perform his Milah (and that of Yishma'el). 4
This is also the opinion of Ibn Ezra, as well as Radak to 17:23.
Avodah Zarah 27a; Targum Yonasan to 17:13. To resolve the opinion of Ramban, see Answer #4. Perhaps Avraham employed numerous people who were born circumcised, or were circumcised through such people. Some hold that females can circumcise males (PF).
Shem ben Noach had been born circumcised
QUESTIONS ON RASHI
Rashi writes: "'On this very day' - [when] Avraham reached 99 years old... he was circumcised." Ramban asks
Gur Aryeh: According to Rebbi Eliezer, the Avos were born in Tishrei; and the Milah and the news of Yitzchak's impending birth, were also in Tishrei. (Refer to 17:21:151:1:1. Yitzchak would be born about six months later, in Nisan.) When Rashi to 18:10 writes that Avraham was informed in Nisan, he follows the opinion of Rebbi Yehoshua (Rosh Hashanah 11a), that the world was created in Nisan, all the Avos were born in Nisan, and so Avraham's Milah was also in Nisan. 1
Rashi writes: "'On this very day' - that Avraham reached 99 years old, and Yishma'el reached 13 years old, they were circumcised." Ramban asks
Gur Aryeh: Prior to Matan Torah, one became liable to punishment at the hands of Heaven only at 100 years old. The Torah tells us that Hashem commanded Avraham in the Mitzvah of Milah on this exact date, for it was not fitting that Avraham should even begin his 100th year without Milah. Yet, even prior to Matan Torah, a minor would leave his father's control at age thirteen. Had Yishma'el reached that age, whether to perform this Mitzvah would have been up to him, and he would never have performed it. 1
Gur Aryeh: The commandment of Milah had not been addressed to Yishma'el personally, but rather to Avraham [and that command included the members of his household].