129:18 If there is a bris on Rosh Hashana, it is performed after reading the Haftarah and before blowing the shofar. There is an allusion to this in the phrase “remember the covenant with Abraham and the binding of Isaac,” referring to bris and shofar, respectively. (Also, circumcision is more common than shofar-blowing and the rule is to perform more-common mitzvos before less-common mitzvos – Mishnah Brurah 584:11.) On Shabbos, the bris is performed after Ashrei. If the bris is to be held at the family’s home, it is performed after shul. (If the family lives very close and if it’s not a huge imposition on the congregation, then it’s still good to circumcise before blowing shofar so that the congregation will have that mitzvah to their merit when they blow – MB ibid.)
129:19 If one has already fulfilled his obligation in shofar-blowing but he needs to blow shofar for others, he may say the brachos again. In any case, it is preferable that the one who still needs to fulfill the mitzvah recites the brachos. If one has already performed the mitzvah and then blows shofar for a woman, he should not recite the brachos again; the woman should do it. This is because, according to halacha, women are not obligated to hear the shofar since they are exempt from positive, time-bound commandments. Some authorities say that one who has already fulfilled his obligation of shofar should not blow for women at all. If one wants to blow for women, he should do so before hearing the shofar in shul. He should recite the brachos over his own shofar blowing and have in mind the intention to fulfill his obligation through this. This should not be done in the first three hours of the day because at that time one is not permitted to blow shofar as an individual. He should either blow shofar for the women at the same time as the shofar is blown in shul or after the shofar has been blown there. If he does so after, then he should have in mind the intention not to fulfill his obligation through the shofar blowing in shul but rather through the blowing that he will do for the women, and he should recite the brachos on that. Even though the one blowing for the women will later go to shul for musaf and he will hear the shofar blowing in the Amidah, he does not have to repeat the brachos because all the shofar blowings are considered a single mitzvah. If the woman is feeling weak and needs to eat before hearing the shofar, she may do so.