Gemara

22b (Mishnah): Anyone can write a Get, even a Cheresh (deaf-mute), lunatic or minor.

Question: Why is this valid? They lack Da'as (understanding)!

Answer #1 (Rav Huna): The case is, an adult supervises them.

Question (Rav Nachman): Would adult supervision enable a Nochri to write a Get?! You cannot say that it does. A Beraisa disqualifies him!

Answer: A Nochri writes for his own motive. (A Cheresh, lunatic or minor writes according to what the adult tells him.)

Retraction (and Answer #2 - Rav Nachman): I erred. The Seifa (23a) says that a Nochri cannot be a Shali'ach to give a Get. This implies that he can write one. The Beraisa is like R. Elazar, who requires a Get to be written Lishmah (for the woman being divorced). The Mishnah is like R. Meir, who does not require this.

Answer #3 (Rav Yehudah): A Cheresh, lunatic or minor may write the Tofes (the standard text common to all Gitin), but one with Da'as must write the Toref (what is particular to this Get, e.g. their names). The Mishnah is like R. Elazar.

Chulin 2a (Mishnah): All may slaughter, and the slaughter is Kosher, except for a Cheresh, lunatic or minor, lest he slaughter improperly. If any of them slaughtered while others looked on, the slaughter is Kosher.

12b (Rava): This is like R. Noson, who does not require intent for Shechitah.

Yevamos 104b (Mishnah): If the Yavam or Yevamah was deaf, or he was a minor, the Chalitzah is Pasul.

Pesachim 38a (Rabah): "U'Shmartem Es ha'Matzos" - Matzah must be guarded for the Mitzvah of eating it (on Seder night).

Rishonim

Rif (Gitin 11b): Even when an adult supervises, the Cheresh, lunatic or child must leave over the Toref, for it must be Lishmah.

Rambam (Hilchos Tefilin 1:11): If a Nochri tanned hide, it is Pasul for a Sefer Torah or Tefilin. He acts for himself, not for the Yisrael who hired him. Therefore a Nochri is Pasul for any action that requires Lishmah.

Rosh (Hilchos Sefer Torah (after Menachos) 3): R. Baruch says that if a Nochri tans hide and a Yisrael oversees, it helps for a Sefer Torah. He does according to the Yisrael more than a Kusi does, for the Kusi thinks that he knows better than the Yisrael. The Ge'onim permitted this, even though regarding a Get, we say that a Nochri acts for himself. The Rambam forbids. People follow R. Baruch, for Yisrael tanners are not found everywhere. This seems correct. The entire Toref of a Get must be written Lishmah, but for tanning it suffices to begin, i.e. to say that he puts the hide into the lime Lishmah. The Nochri finishes according to the Yisrael's command. Even R. Yehudah, who requires Milah Lishmah, allows a Nochri for Milah (Avodah Zarah 26b-27a). We do not say that the Nochri acts for himself. Since it is done in a moment, surely he does according to what the Yisrael tells him.

Poskim

Shulchan Aruch (OC 460:1): We do not knead Matzos for the Mitzvah through a Nochri, Cheresh, lunatic or child.

Rashba (1:26, brought in Beis Yosef (OC 460 DH Ein): Matzah must be guarded, therefore one is not Yotzei with Matzah kneaded by a Nochri, Cheresh, lunatic or child. Adult supervision does not help for a Nochri, for he acts for himself. A Cheresh, lunatic or child can write a Get only if he is supervised. Supervision does not help for Shechitah, according to Chachamim who require intent. Tosfos (12b DH Man) says that others see, but do not teach him and admonish him to intend, so it is unlike 'Omed Al Gabav' that helps for a Get. However, this does not distinguish a lunatic from a Cheresh or child. Regarding Chalitzah, we distinguish. Chalitzah of a deaf or minor Yavam is Pasul and forbids Yibum. We do not say that Chalitzah of a lunatic forbids. Tosfos says that a lunatic has no Da'as at all; supervision does not help at all if there is no indication (of intent) in the matter he does, e.g. Shechitah and Chalitzah. Regarding a Get, they tell him to write for Ploni and Plonis, and he does. He musters up intent and writes Lishman. According to this, if a Cheresh or child, who has a little Da'as, is supervised, he can knead for Matzah and we are Yotzei with it. One is not Yotzei with a lunatic's Matzah.

Rashba (ibid.): R. Yonah says that for anything that can be done through a Shali'ach, such as Shechitah or writing a Get, intent of a supervisor (David) helps like intent of the one doing it (Levi). It is as if Levi is David's Shali'ach. Therefore, a lunatic can slaughter or write a Get if a Pike'ach (healthy person) supervises. Shelichus does not help for Chalitzah. Therefore, a Cheresh, lunatic or child cannot do Chalitzah, even with supervision. According to this, the same applies to kneading l'Shem Matzas Mitzvah. It seems to me that supervision helps only when there is an indication that he acts Lishmah. Writing a Get with the proper names is an indication, so even a lunatic can it when supervised. There is no indication in Shechitah or kneading, so supervision cannot help. There is a small indication in Chalitzah, i.e. the supervisors instruct them, therefore it helps for a Cheresh or child, but not for a lunatic, who has no Da'as.

Magen Avraham: The Rosh (Hilchos Sefer Torah 3) holds that supervision helps Therefore, I agree with the Bach, who says that if the only Matzah available was made by a Nochri supervised by a Yisrael, one may use it. It is good if the Yisrael helps a bit. It is better to use a minor than to use a Nochri.

Gra (1): The Rif says that adult supervision helps only for the Tofes, but not for the Toref, which needs Lishmah mid'Oraisa. All the more so one may not use a Nochri (for Matzah), who may not write even the Tofes.

Mishnah Berurah (1): Some are lenient to allow minors to make holes in the dough, since adults supervise, and more helpers speed the process.

Mishnah Berurah (2): Matzah for the rest of Pesach merely needs to be guarded from becoming Chametz. One may rely on minors with adult supervision. Yisrael are stringent to require l'Shem Mitzvah for all Matzos.

Mishnah Berurah (3): Those who permit relying on supervision require constantly exhorting. It is not enough to tell them once.

Mishnah Berurah (4): Pouring water on the flour is part of kneading. Those who allow minors to do this do not fulfill the Mitzvah.

Kaf ha'Chayim (8): If minors kneaded and adults saw from afar, one may eat it according to the Bach, who holds that guarding is mid'Rabanan. One may not bless 'Al Achilas Matzah', perhaps even if adults supervised. If he helped a bit and told him to do l'Shem Mitzvah, surely one may bless.