A. Leather Shoes:
One may not place leather shoes on a child even if the child is below the age of education. Certainly it is forbidden to do so if the child is above the age of education. If one’s child wore the shoes on his own, then if he is below the age of education, there is no need to remove the shoes from his feet or tell him to remove it. If however the child is above the age of education, the father must protest his son from doing so.
Children are not to be bathed on Yom Kippur, just as is the law by an adult, unless they are dirty, in which case one may wash off the dirty area.
Children are not to be anointed on Yom Kippur, just as is the law by an adult.
Below age of Chinuch: Children who are below the age of education [under 9-10 years old] may be fed as normal, and it is even forbidden to delay their meals from their regular times, being this can lead to danger. Even if the child desires to fast, he is to be protested and forced to eat. Nevertheless, a child may only be fed for his own sake and not for the sake of fulfilling a Mitzvah. [The above is from the letter of the law, however many are accustomed to permit children to fast for a certain amount of hours even if they are below the age of nine. It is however forbidden to force them to fast even for a few hours, even at night. Thus, if they ask to eat or drink they are to be allowed.]
9-10 Years old: Beginning from the age of nine, for a healthy child, and ten, for a weak child, both boys and girls are to be educated to fast on Yom Kippur, through having their meals delayed from their set time. One is to delay the meal an hour or more from its set time based on the amount of time delay the child can handle. Thus, if the meal is normally eaten at 3:00 pm they are to eat it at 4:00 pm or later. [However, if the child is very thirsty, he may be given to drink even at night.]
11 Years old: It is disputed whether a child above the age of 11 must fast on Yom Kippur. Practically, although the stringent opinion is the main Halachic ruling, nevertheless, one may be lenient in a case that the child is too weak to continue fasting, even if there is no danger involved. Based on this, people today are no longer accustomed to educate their child to fast the entire day of Yom Kippur even after 11 years old, until they become Bar or Bas Mitzvah, as today all children are considered weak. Nevertheless, if one knows for certain that the child is healthy and strong enough to sustain the fast, then this leniency does not apply. [It is proper that the child fast until at least midday.]
From Bar and Bas Mitzvah: When a child has reached the age of 12 years old for a girl and 13 years old for a boy, then if he or she has grown two pubic hairs, they are considered adults and are obligated in all the commands, including the fasts. If the children have reached this age, but do not have two pubic hairs, they must nevertheless guard all the commands out of doubt that perhaps the hairs already grew and later fell off.
Obligation of the father to reprimand his child for not fasting: The obligation of educating a child in the above is on the father. Thus if the father sees his child disobeying the above he must reprimand him.
Yaleh Veyavo in Birchas Hamazon: A child who eats a Kezayis of bread on Yom Kippur, is to recite Birchas Hamazon. In Birchas Hamazon, one is to add Yaleh Veyavo prior to Uvinei Yerushalayim and say “Beyom Hakippurim Hazeh”. [However, some Poskim rule that based on Admur in the Siddur one is not to recite Yaleh Veyavo in Birchas Hamazon. Practically it is to be recited.] When Yom Kippur falls on Shabbos he is to also add Ritzei.
If a child eats bread on Yom Kippur, is he to wash hands?
Yes, the hands are to be washed up until the wrists as is normally done during the year.
The first Yom Kippur fast of the Rebbe Rayatz:
Rabbi Yosef Yitzchak of Lubavitch related: When I was seven years old, my father [Rabbi Shalom Dov Ber of Lubavitch] said to me on the eve of Yom Kippur: “In the evening and all night it is forbidden for you to eat. Tomorrow in the morning until noon you do not need to eat. From then on it depends on your will.” My father explained to me the gravity of eating on Yom Kippur, and concluded: “If you want to eat, do not ask anybody, only come to me. I have prepared food, water, and juice for you. If I am in the middle of the Amida prayer, wait for me; just be careful not to ask for food from anyone else.” That year I completed the fast for the first time in my life, and the year after that it was already an easy matter.
 Admur chapter 616
Background: It is forbidden to cause a child to transgress any command, even Rabbinical, and even if the child is below the age of education, nevertheless, one may not cause him to transgress a Mitzvah. [Admur 343/5] Nevertheless, those commands that involve oppression were never fully decreed on children. Therefore, in general it is permitted for an adult to give a child an item which is prohibited due to oppression.
 Admur 616/2; See Piskeiy Teshuvos 616/1
 Being that a child receives no oppression if he does not wear leather shoes, it is therefore forbidden for an adult to place it on him, just like it is forbidden for an adult to feed a child non-kosher food irrelevant of the child’s age.
Other opinions: Some rule that if the child is below the age of understanding, then one may place leather shoes on them. [Piskeiy Teshuvos 616/1]
 Admur 616/4 and 7
 Admur 616/2
 Admur 616/4
 Admur 616/3
Background: From the letter of the law it is permitted to bathe a child who is below the age of Chinuch with cold water. Likewise, one may bathe minority of his limbs with hot water that was heated before Yom Kippur. This is because the laws of oppression were not decreed against children below the age of Chinuch [Admur 616/1] However, today that children are no longer bathed daily, and it is hence not considered an oppression to avoid bathing for one day, it is therefore not allowed to bathe them on Yom Kippur even if the child is below the age of Chinuch, and so is the custom. [Admur 616/3] It is certainly forbidden to bathe a child who is above the age of Chinuch, and if the child does so on his own, the father must protest his actions. [Admur 616/4 and 7]
Other opinions: Some Poskim rule it is permitted to bathe a child in hot water on Shabbos/Yom Kippur. [Bach, brought in M”A ibid] The M”A ibid negates his opinion and so rules Admur.
 Admur 616/3
Background: From the letter of the law it is permitted to anoint a child who is below the age of Chinuch. This is because the laws of oppression were not decreed against children below the age of Chinuch [Admur 616/1] However, today that children are no longer anointed daily, and it is hence not considered an oppression to avoid anointing for one day, it is therefore not allowed to anoint them on Yom Kippur, even if the child is below the age of Chinuch, and so is the custom. [Admur 616/3] It is certainly forbidden to anoint a child which is above the age of Chinuch and if the child does so on his own the father must protest his actions. [Admur 616/4 and 7]
 see Piskeiy Teshuvos 616/2
 Admur 616/10
 Meaning, one does not have to delay their meals. [so is implied from Admur ibid as otherwise a child before nine and over nine would have the same law. Hence, one must learn that a child before nine is not oppressed at all even to delay a meal.]
 As is implied from the ruling that if the child desires to fast [or delay the meals] he is to be protested. Thus, certainly one may not force the child to fast. [ibid]
 Vetzaruch Iyun why a 2/3 hour delay would lead to danger?
 See also the laws of Tisha B’av chapter 556 Giving a child to drink from the wine is also not done, being that children may only be given to eat on Tishe Beav for their own sake, in order not to accustom them to eating on Tishe Beav even when they get older. [M”A 556] Thus we see that even a child who may be fed, may only be fed for his own sake.
 Ashel Avraham Butchach 616; Chikrei Lev 109; See Piskeiy Teshuvos 616/2
 Alef Hamagen 616/2
 Admur 616/5
 Yalkut Yosef brought in Piskeiy Teshuvos 616/2
 Admur 616/6 and 11
 Some Poskim rule that when the child reaches 11 years old, he must begin to fast for the entire day. This obligation is a Rabbinical decree, and applies to both males and females [1st opinion in Admur ibid; Rambam] Other Poskim however rule that until the child reaches Bar/Bas Mitzvah, there is no obligation for him/her to fast the entire day, and they only are obligated to fast by delaying their meals.
 Admur 616/6
 Admur 616/11
The reason: 1) Today the children learn Torah, and the learning of Torah weakens them [We find this concept also regarding the Onah obligation that a Torah Scholar need only have relations once a week, being that his strength is lessened due to his Torah learning] and 2) Today people are weaker than previous times being that “weakness has descended upon the world, and thus a regular child above 11 is not considered strong enough to fast. [Admur ibid]
 So is implied from end of Admur 616/11; and so rules Mishneh Berurah and Piskeiy Teshuvos 616/2
 Yalkut Yosef brought in Piskeiy Teshuvos 616/2
 Admur 616/8
 Admur 616/4
 This applies to all laws which a child is required to be educated in, as explained in chapter 343.
 Admur 618/18; first opinion in 188/11
Other opinions: Taz 618/10
 Shaar Hakolel 34/9
 Piskeiy Hasiddur 185; See however Hiskashrus who writes it is to be recited in Harachaman, See Nitei Gavriel Bein Hametzarim 66/11
 Levushei Mordechai Tinyana Y.D. 11; Yad Sofer 7; Shevet Halevi 8/139; SSH”K 39/31; Lehoros Nasan 2/42; Shraga Hameir 5/45; Piskeiy Teshuvos 613/2; 618/15
 Likkutei Dibburim, vol. 4, p. 1418