Gemara

Mishnah: We read Parashas Zachor on the second Shabbos and Parashas Parah the third.

18a - Mishnah: If one read the Megilah by heart, he was not Yotzei.

Rava: We learn this from a Gezeirah Shavah Zechirah-Zechirah from remembering Amalek, which must be in a Sefer.

Question: How do we know that Zechirah regarding Amalek is reciting it? Perhaps it suffices to think about it!

Answer: "Remember, don't forget" - forgetting is in the heart, so "remember" must come to obligate reciting it.

Rishonim

Rosh (Berachos 7:20): R. Eliezer freed his slave to make a Minyan (Berachos 47b). It was to enable saying Kedushah or Borchui - "V'Nikdashti b'Soch Benei Yisrael". It is difficult to say that it was Parashas Zachor, on which Kri'as ha'Torah is mid'Oraisa.

Sefer ha'Chinuch (Mitzvah 603): We must recite Ma'ase Amalek; it does not suffice to remember in the heart. The Torah did not specify how often we must do so. It seems that once in one or two or three years is enough. Everywhere, Benei Yisrael read the entire Torah in this time, and are Yotzei through this. It is our custom to fix a time for this Mitzvah, i.e. the Shabbos before Purim. The Mitzvah applies only to men, for they take vengeance in war.

Rebuttal #1 (Minchas Chinuch DH veha'Rav): Sefer ha'Chinuch seems to exempt women also from the Lav. It appears that they are obligated even in the Aseh, since it is not Zeman Gerama. It is not clear that the Mitzvah depends on avenging Ma'ase Amalek. And even if it is, regarding Milchamas Mitzvah even a Kalah leaves the Chupah (to fight - Sotah 44b).

Rebuttal #2 (Sedei Chemed Ma'arachas Zayin Kelal 13 DH v'Im): Why should women be exempt? Women were instrumental in the victories against Haman, the Yevanim and Sisera! The Chinuch himself (Mitzvah 425) obligates women in the war against Amalek!

Note: In Mitzvah 425 the Chinuch obligates women to kill the seven Kena'ani nations. In Mitzvah 603, the Chinuch obligates males to wipe out Amalek. The Rambam (Aseh Shemoneh Esre) obligates also females to wipe out Amalek

Defense (Radvaz Hilchos Melachim 7:4): A Kalah leaves the Chupah, i.e. she forfeits the week of Sheva Berachos because her Chasan must go to fight. Alternatively, women supply water and food for their husbands, like is done nowadays.

Avnei Nezer (OC 509:2 DH veha'Nir'eh): The Mitzvah to kill Amalek is no more stringent than other Misos Beis Din, which are not Docheh Shabbos. The Heter to fight wars on Shabbos does not apply. Therefore, it is Zman Grama so women are exempt.

Griz (Shmuel 1:15:1, printed with Beis ha'Levi Al ha'Torah): Hash-m commands through a Navi when Yisrael should destroy Amalek - "Milchamah la'Shem ba'Amalek". An individual who sees an Amaleki is not commanded to kill him.

Ramban (Devarim 25:17): We expound that one must remember Ma'ase Amalek by saying (the Parashah). Perhaps this obligates reading the Parashah in a Minyan, and this is the source for Parashas Zachor, and an Asmachta for Kri'as ha'Megilah. It is better to say that the Torah commands not to forget Amalek until we destroy Amalek, which necessitates telling future generations.

Poskim

Shulchan Aruch (685:7): People who live in settlements who do not have a Minyan must go to a place with a Minyan on the Shabbosos of Parashiyos Zachor and Parah to hear the Parashiyos.

Terumas ha'Deshen (108): Even though Kri'as ha'Megilah overrides (virtually) all (positive) Mitzvos, one must be even more zealous to hear Parashas Zachor. Most say that one may hear the Megilah without a Minyan, but Parashas Zachor requires a Minyan, and it overrides the Lav of "l'Olam Bahem Ta'avodu." If one cannot fulfill both, it is better to hear Parashas Zavhor in a Minyan.

Magen Avraham (DH bi'SHLaH): People are not so zealous about Parashas Zachor because the Torah did not specify when it must be read. Chachamim enacted to read it before Purim, when we read "Va'Yavo Amalek". If one cannot fulfill both, he should hear the Megilah in a Minyan on Purim. By hearing "Va'Yavo Amalek" he remembers what Amalek did, and is Yotzei b'Di'eved. Nevertheless one should be zealous to hear it in a Minyan.

Question (Mishnah Berurah 16): Parashas Zachor commands us to remember "Asher Korcha ba'Derech va'Yzanev Becha", tell this to our children, and wipe out Amalek's name. This is not mentioned in Va'Yavo Amalek!

Kaf ha'Chayim (34): It is better that they go to hear Parashas Zachor, which is first; perhaps they will (unexpectedly) be able to hear also the Megilah. Also, we do not sour the opportunity to do a Mitzvah.

Mishnah Berurah (16): Even if one has a Sefer Torah, and even if the Chiyuv to hear the Parashah with a Minyan is mid'Rabanan, since the Ikar Mitzvah is mid'Oraisa one must hear it with a Minyan.

Mishnah Berurah (15): Many say that Parah is mid'Rabanan. Nevertheless, one must have intent to be Yotzei.

Kaf ha'Chayim (29): We do not bless on the Mitzvah to remember Amalek because we do not bless on destruction, even of Nochrim, just like Hash-m did not allow the angels to sing on the night of Kri'as Yam Suf.

Kaf ha'Chayim (30): Women are exempt from hearing Parashas Zachor. However, they are obligated to remember Ma'aseh Amalek. If they hear the Parashah they are rewarded like one who did a Mitzvah he was not obligated to do.

Sedei Chemed (ibid., DH Gam): Surely, the Chiyuv to hear it in a Minyan from a Sefer Torah is mid'Rabanan. Mid'Oraisa one may recite it himself.

Hagahah (to Sedei Chemed): This is unlike the Terumas ha'Deshen (106, brought above).

Rema: If they cannot hear the Parashiyos, they should at least read them with the proper cantillation.

Kaf ha'Chayim (35): We read it with a Berachah only from a Sefer Torah in a Minyan.

Mishnah Berurah (18): Opinions differ about whether we read "Zeicher Amalek" with a Tzeirei, or 'Zecher' with a Segol. The reader should read it both ways.

Question (Chasam Sofer EH Teshuvah 119): We are commanded not to forget Ma'ase Amalek. The Shi'ur for forgetting is one year. In Eretz Yisrael, they used to finish the Torah only once every three years. If so, once every three years they did not read (about Amalek) in Parashas Beshalach or in Ki Setzei. If the year was Me'uberes (so Parashas Zachor was 13 months after the previous Parashas Zachor), did they transgress?!

Answer (Chasam Sofer ibid.): The Ramban says that only one year is allowed to redeem a Bayis Ir Chomah, for after this it is forgotten. When the year is Me'uberes, one may redeem in the extra month (Erchin 31b), i.e. one does not forget until a full year including the extra month.

Note: The Chasam Sofer explains the custom to finish the Torah in three years according to Rebbi. Since Chachamim allow only 365 days to redeem even in a leap year, this is the Shi'ur for forgetting. Even according to Rebbi, it is not clear what was done in a leap year if Parashas Zachor was read earlier in Adar (e.g. the ninth or 10th) the previous year, and later (e.g. the 12th or 13th) in Adar Sheni of the next year. I did not find anyone who discusses how the Mitzvah was fulfilled before the Nes of Purim.

Rema (282:4): A minor may read the Maftir of Korbanos Musaf or during the four Parashiyos. Even though some disagree, this is the custom.

Gra (DH Af): The Rosh (3:5) says that it is unreasonable that a child called for Maftir, who is not among the seven required Aliyos, can read something that must be read that day.

Bi'ur Halachah (DH O b'Arba): If the child can read word for word with the Shali'ach Tzibur, it should be permitted. Nevertheless, some are stringent about Parashiyos Zachor and Parah. Perach Shushan is stringent about all four Parashiyos.

Teshuvas Binyan Tziyon (ha'Chadashos 8 DH Teshuvah): I have a tradition from R. Noson Adler that women are obligated. It seems that this is because Parashas Zachor is not Zman Grama. It may be read any time during the year.

Suggestion: The Rosh (ibid.) suggests that perhaps R. Eliezer freed his slave in order to read Parashas Zachor. If a slave was obligated, he would join for the Minyan to read it! This shows that slaves are exempt; the Halachah for women is the same as for slaves.

Rejection (Binyan Tziyon ibid.): We learn from "V'Nikdashti b'Soch Benei Yisrael" that a Minyan of Benei Yisrael is required.

Kaf ha'Chayim (40): It is better to call an adult for Parashas Zachor. If a minor was called he does not step down. However, an adult Shali'ach Tzibur must read it out loud.

Other applications

To be concerned for the opinion of the Chasam Sofer (ibid.), in a leap year one should intend to be Yotzei Zechiras Amalek when hearing those same verses in Parashas Ki Setzei. The Chasam Sofer says that one can be Yotzei through Parashas Beshalach, but the Mishnah Berurah (16) challenges this. Perhaps the Chasam Sofer agrees; he asked that even if one could be Yotzei also through Parashas Beshalach, there is a difficulty.