Women are not obligated to hear Shofar. Nevertheless if they desire to blow Shofar they may do so with a blessing. Practically women today have accepted the hearing of Shofar as an obligation and hence they must do so due to this custom.
May a man blow Shofar for women? It is permitted for a man to blow Shofar for women even if he has already fulfilled his obligation. It is even permitted for him to carry a Shofar through a public domain for the sake of women hearing Shofar. However if the man has already fulfilled his obligation of hearing Shofar [or plans to do so with a later blowing] then he may not say the blessing for a woman, and rather the blessing is to be said by one of the women. If a man desires to blow with a blessing for women then he is to blow for them before hearing Shofar in Shul and hence fulfill his obligation with this blowing. Alternatively he can intend not to fulfill his obligation with the blowing in Shul [of Meyushav] and then blow for women with a blessing [between Meyushav and Meumad] having intent to fulfill his obligation with this blowing.
May a woman blow Shofar for another man to fulfill his obligation?
No, as will be explained in H!
May a woman blow Shofar for herself or for another woman to fulfill the Mitzvah?
How many blows is a woman to hear?
A woman only needs to hear the 30 sounds required to fulfill the Mitzvah [as explained in Chapter 3]. Nevertheless it is proper for them to also hear the blows of Musaf and not speak until its conclusion.
Must a woman perform Hataras Nedarim if she is unable to hear Shofar one year for whatever reason?
 588/6; 589/2; Michaber 589/3; Kedushin 29
The reason: The reason for this is because hearing Shofar is a Mitzvas Aseh Shehazman Grama. [ibid] Regarding a Tumtum and Androgenus see 588/9.
Admur 589/2; Michaber 589/6; R”H 33a as rules Rebbe Yossi and Rebbe Shimon; Rabbeinu Tam; Ran; Rosh there; Beis Yosef 589; See Kaf Hachaim 589/22
The reason it is allowed for women to blow Shofar on Rosh Hashanah despite it being a Shvus: Although blowing Shofar on Shabbos or Yom Tov is forbidden if it is not done for the sake of the Mitzvah and women are not obligated in the Mitzvah nevertheless the Sages allowed women to blow it if they intend to do so for the sake of the Mitzvah. The reason for this is because this specific prohibition of blowing a Shofar is not an actual complete Shvus as it is merely forbidden due to Uvdin Dechol, thus for the sake of appeasing the women the Sages permitted this light Rabbinical matter. [589/2]
 589/2; Rama 589/6; Rabbeinu Tam R”H 33a; Ran; Rosh in name of Mahritz Geios; Tur 589; Ritva Sukkah 2; Rashba 123; Maggid Mishneh Shofar 2 in name of Rashba; Shut Min Hashamayim 1; Birkeiy Yosef 654/2; Yosef Ometz 82; See Kaf Hachaim 17/4; 589/23
The reason it is allowed for women to say a blessing despite them not being obligated in the command: Women are accustomed to say a blessing on even those commands that they are exempt from due to it being a Mitzvas Aseh Shehazman Grama. Thus in this case as well women may say the blessing before blowing Shofar. [ibid] In 17/3 Admur explains that women may say a blessing of “Asher Kidishana … Vitzivanu..” on commands that they are exempt from fulfilling being that the men are commanded in this Mitzvah. Furthermore although they do not receive as much reward as a man who is commanded in the Mitzvah, they do receive some reward and thus they too may say a blessing just like a man. [17/3]
Other Opinions-Sefaradim: Some Poskim rule that women may never recite a blessing over any Mitzvah that they are not obligated in. [Michaber 589/6; Beis Yosef 589; Rambam Hilchos Tzitzis; Igur in name of Reb Yeshaya; Rashi, brought in Hagahos on Rambam] The Michaber ibid rules that they cannot say a blessing as Safek Brachos Lihakel. [Beis Yosef 589] The Birkeiy Yosef ibid states that after seeing the answer from heaven in Shut Min hashamayim he retracted his ruling like the Michaber and would tell women to recite a blessing. He concludes that if the Michaber would have seen the Shut Min Hashamayim certainly he too would have retracted his ruling.
Admur 17/3; Shut Rav Akiva Eiger 1 Hashmatos; See Ashel Avraham Butchach 689 and Mahadurah Tinyana 592; so is implied also from Kitzur SHU”A 129/19; Ateres Zekeinim 689
 589/2; Michaber 589/6
Admur ibid; Michaber ibid; Rama ibid; Ravaya brought in Rosh R”H 33; Tur; Beis Yosef; Levush; Erech Hashulchan 589/3; M”B 589/8
Other Poskim: Some Poskim rule it is forbidden for men to blow the Shofar on behalf of only women as it is forbidden to blow the Shofar for no need. [Baal Haitur brought in Tur 589; Peri Chadash; See M”E 589/12] The Kaf Hachaim 589/27 concludes that since there are Achronim that agree to the ruling of the Peri Chadash the custom therefore is that women come to Shul to hear Shofar and only in a case of need do we blow for them outside of Shul.
Admur 589/2; Ravaya brought in Rosh R”H 33; Tur; Beis Yosef; Chida in Yosef Ometz 82; Kaf Hachaim 589/28; Igros Moshe 3/94; Az Nidbaru 13/38 [permitted even for female children]; Ketzei Hamateh 589/5 [“so is the custom”]; Piskeiy Teshuvos 518/4; 589/5
Other Opinions: Some Poskim rule one may not carry the Shofar through a public domain that does not contain an Eruv simply for the sake of blowing for women. [Shaagas Aryeh 106 brought in Machazik Bracha 589/5; Shaareiy Teshuvah 589] The Chida in Yosef Ometz ibid negates this opinion and concludes like above. See Piskeiy Teshuvos ibid footnote 16 that perhaps today it is permitted according to all opinions as women have accepted upon themselves to hear Shofar and it is hence like a vow which is Biblical and one may hence carry on their behalf according to all.]
The reason why a man may not say the blessing for women: Women are not obligated in the command of Shofar and hence are not commanded to say the blessing. Thus a man has no obligation to recite the blessing for them to fulfill this optional Mitzvah. In such a case if a man says the blessing for women it is considered a needless blessing and is hence defined as a blessing in vain. However women themselves can choose to say the blessing for reasons explained in previous footnotes. [ibid] Vetzaruch Iyun as to how to understand this reason. Why can’t men say this optional blessing for women just like women can say it for themselves? In other words if we permit an optional blessing for women, why can’t we also permit it for men to say for the sake of women? Perhaps the reason is because since the blessing is optional and the man gains nothing for saying it [no even an exemption from Areivus] it is therefore considered an unnecessary blessing for him to say. However women may say it being that she gains the fulfillment of the Mitzvah and hence it is not unnecessary.
 This means that one is to leave Shul to blow for women between the first and second sets of blowing [between the blowing before Musaf and Musaf] and then return to Shul to hear the blowing of Musaf [Meumad]. [However one is not to have in mind not to fulfill his obligation and then only blow for women after Musaf as in such a case he has lost out on the Mitzvah of Tekios Meumad in Musaf which is only done with a Minyan.] Now although one makes an interval between his blessing and the blows of Musaf being that he says the blessing in the house of the women and returns to Shul for the blowing of Musaf, nevertheless he is not required to repeat the blessing before Tekios Meumad of Musaf as even if one talked in-between he is not required to repeat the blessing as explained in 592/6. [ibid]
Admur ibid; See M”E 589/12 that says this is the best option in order to suspect for those that rule one may not blow on behalf of women. However see Alef Hamagen 589/11 that negates this as we do not ask of one to exclude himself from the community Mitzvah.
 M”E 589/8; Kaf Hachaim 589/15
Shaagas Aryeh 105; Machazik Bracha 589/4; Shaareiy Teshuvah 589; Kaf Hachaim 589/26
Piskeiy Teshuvos 589/2
Piskeiy Teshuvos 589/4
 M”A 581/12; Degul MerivavaYoreh Deah 214/2; Yabia Omer 2/30
Kaf Hachaim 568/46; 581/71; 589/34; Ben Ish Chaiy Netzavim 17; see Shach 214/2