Gemara

Beraisa: All count towards the seven, even a child or woman, but Chachamim said that a woman should not read for it is dishonorable to the Tzibur.

24a - Mishnah: A child may read the Torah and translate.

Rishonim

Rif and Rosh (3:5) bring the Beraisa and Mishnah.

Ran (DH ha'Kol): A child or woman may complete the seven, but all seven (Ran 15a Sof DH Masnisin - the majority) may not be children or women. When only the first and last Olim bless, these Aliyos must be given to men, for men cannot be Yotzei with the Berachah of a child or woman. Nowadays, everyone blesses, so a child or woman may read first or last.

Rosh (Kidushin 1:49): R. Tam says that a woman may bless on a Mitzvah from which she is exempt. R. Yitzchak bar Yehudah proves this from the Heter for women to read the Torah. R. Tam rejects the proof, for even if the Olah already blessed for his Mitzvah of Talmud Torah, he blesses again for Kri'as ha'Torah (i.e. the Berachah is to honor the Kri'ah). Also, perhaps the Heter for a woman to read was when only the first and last Olim used to bless, and she would be Olah in the middle (and not bless).

Rambam (Hilchos Tefilah 12:17): A woman should not read b'Tzibur for it is dishonorable to the Tzibur. If a child knows how to read and knows Whom we bless, he counts towards the number that must read.

Source (Hagahos Maimoniyos 70): In the Yerushalmi, Amora'im argue about whether a child must know how to bless or Whom we bless in order to count towards the seven on Shabbos. Likewise, when three read a child may complete the count.

Pirush ha'Mishnayos (24a): A Ga'on says that a child may read only after three have read.

Poskim

Shulchan Aruch (OC 282:3): All count towards the seven, even a child or woman, but Chachamim said that a woman should not read for it is dishonorable to the Tzibur.

Beis Yosef (DH v'Chosav ha'Rav): The Avudraham says that in a city of all Kohanim a Kohen reads the first two Aliyos, and children and women read the rest. The Roke'ach does not allow a child to be one of the three, but he may be one of seven, i.e. after three have read.

Note: In Sof Siman 135 (DH Kosav ha'Roke'ach), the Beis Yosef says that the Roke'ach allows a child to have an additional Aliyah (which is forbidden when three read), but not an obligatory Aliyah.

Prishah (3): The Tur holds like the Rambam who forbids women to bless on Mitzvos from which they are exempt. Therefore, a woman or child may not receive an Aliyah nowadays, when every Oleh blesses.

Question: It is the practice to allow children to receive Maftir!

Answer #1 Prishah (3): The initial enactment was to allow children to receive Maftir. If so, it was enacted that they may bless for Kavod ha'Torah.

Answer #2 (Birkei Yosef 7, brought in Kaf ha'Chayim 25): A child may receive an Aliyah nowadays, for Talmud Torah applies to him, and others are commanded to teach him.

Erech Lechem (135:13): The Rambam says that a child counts towards the number that must read, i.e. even when three read.

Mishnah Berurah (11): A child may read only when seven read, but not when three, four, five or six read.

Mishnah Berurah (12): Some say that if the only Kohen is a child we call him (to read). However, the Magen Avraham (6) learns from "V'Kidashto Ki Es Lechem Elokecha Hu Makriv" that we are commanded to honor only adult Kohanim, for only they do Avodah. Therefore, a Yisrael reads instead. Nowadays the custom is to call children only for Maftir, even after seven have read.

Mishnah Berurah (13): A child may not read out loud from the Torah to be Motzi the Tzibur until he brings two hairs. We rely on the Chazakah that a 13 year old brought two hairs. If there is no adult who can read it is better that a child read than to Mevatel Kri'as ha'Torah.

Rema: They can join to be part of the seven, but all may not be children or women.

Source (Gra DH v'Elu): The Mishnah says that a child or woman (singular) counts, i.e. only one. The Poskim's text reads 'All can complete the count...'

Mishnah Berurah (14): 'All', i.e. the majority, may not be children or women.

Note: The Mishnah Berurah explains the Rema. Above, the Mishnah Berurah said that our custom is to call a child only for Maftir.

Rema: A Kena'ani slave is like a woman; if his mother is a Yisraelis he may read.

Hagahos Maimoniyos (60): If his mother is a Yisraelis he is permitted (like a regular Yisrael), because the child of a slave or Nochri with a Yisraelis is Kosher.

Note: Even though the Rema cites Hagahos Maimoniyos, it seems that he argues. Hagahos Maimoniyos connotes that if the child of a slave or Nochri from a Yisraelis would be a Mamzer, he could not read. The Rema explicitly permits a Mamzer to read!