Children joining a Minyan:[1]
Children below the age of Mitzvos do not join a Minyan. However, if there are only 9 people in a Shul and there is a male child present who is of age six and above and knows to whom he is Davening, then some Poskim allow the child to join the Minyan if he holds a Sefer Torah, or Chumash, or even if he does not hold anything in his hand. Practically, the main opinion follows the stringent opinion that a child can never join a Minyan even if he is the 10th person and holds onto a Sefer Torah in his hand. This applies even in a time of need. Nevertheless, those who are lenient like the former approach are not to be protested as they have upon whom to rely, however, even they should only recite the obligatory Kaddeishim and Barchu on this basis. The Chabad custom, however, is not to rely on this opinion, and hence we do not join a child to a Minyan in any circumstance.
Children Forming Their Own Minyan:[2] Children below the age of Mitzvos may not form their own minyan to recite Kaddish, Kedusha, or Barchu, and adults may not assist them in doing so, as this falls under the prohibition of causing a child to transgress (issur sefiyah). Even if children attempt to do so on their own, they should be prevented for the sake of proper chinuch. Furthermore, an adult who hears such a minyan may not respond to their Kaddish, Kedusha, or Barchu.
May a Child Lead the Prayer or Recite on Behalf of the Congregation?[3] A child below the age of Mitzvos may not serve as the chazzan, as one who is not fully obligated in a mitzvah cannot fulfill the obligation on behalf of others. This applies even for Maariv.
Kaddish Recited by a Minor Orphan:[4] A son who is below Bar Mitzvah—and even below the age of chinuch—may recite Kaddish on behalf of a deceased parent. On the contrary, Kaddish was specifically instituted for orphans, including those below the age of mitzvos. If the child is too young to recite it independently, an adult should assist him by reciting it together with him, allowing the child to repeat after him.
[1] See Admur 55:5; Michaber 55:4; M”B 55:24; Shulchan Menachem 1:65; Piskeiy Teshuvos 55:9; Chinuch Habanim Kehilchaso Chapter 17:1-35
[2] Igros Moshe O.C. 2:98; Chikrei Halachos p. 145; Chinuch Habanim Kehilchaso Chapter 17:34
[3] Admur 53:4 and 9 and 13; Michaber 53:6; Rambam Tefila 8:11; Mishneh Megillah 24a; Rosh Yosef Megillah 24a; M”B 55:32; Chinuch Habanim Kehilchaso Chapter 18:1-19; 19:1-27
[4] Rama 376:4; Ashel Avraham 132; See Igros Kodesh 18:266 regarding Mishnayos and Kaddish Derabanan; Pnei Baruch 34:22; Nitei Gavriel 40:7; Piskeiy Teshuvos 132:21; Chinuch Habanim Kehilchaso Chapter 18:13-15