Does one need to repent for sins that he did as a child?
A child [below Bar/Bas Mitzvah] who hit his father or committed other sins while still a child, although he does not need to repent when he grows older, nevertheless, it is proper for him to accept upon himself a certain matter for the sake of repentance and atonement, even though he had transgressed before he became of punishable age. [This applies whether to sins between man and his fellow, or sins between man and Hashem. The purpose of this repentance is not for the sake of removing Divine retribution, being that he is not liable for any punishment. However, the forbidden act, despite not being punishable, leaves a damaging impact and spiritual stain on the soul of the child. It is for this reason that the child should do some act of repentance for the sake of cleansing his soul of any spiritual blemish.] Regarding monetary matters, this will be explained in Choshen Mishpat chapter 349.
Is a child that is below the age of Chinuch encouraged to perform a certain matter for the sake of repentance?
Some Poskim rule that the above Teshuvah is only relevant for children who damaged when they were of age to remember what they did. If, however, they were of very young age, and they don’t remember the incident, they are not required to do Teshuvah at all.
Is one punished in heaven for sins he committed when below Bar Mitzvah?
Some Poskim rule that a child who has reached the age of understanding what is a Mitzvah and Aveira, Onas Hapeutos, is punished for his sins. Veztaruch Iyun Gadol from all the above sources that imply he is not punished.
 Admur 343:11; Rama 343; Mahariy 62; Bach 343; Shvus Yaakov 1:177 brought in Pischeiy Teshuvah C.M. 349:2; Based on story in Shabbos 56b; Baba Kama 98b, brought in Kaf Hachaim 343:32
Other opinions: Some Poskim rule that a child is not required to repent at all for his wrongdoing even when he becomes older. [Beis Yaakov 3, brought in Shvus Yaakov ibid] Practically, we do not rule like his opinion. [Shvus Yaakov ibid; Kaf Hachaim ibid]
 Elya Raba 343:4; Kaf Hachaim 343:33; M”B 343:9
 Bach 343: The verse in Mishlei 19:2 states that “Even without knowledge, a soul is no good.” This teaches us that even a child that transgresses a sin without knowledge causes impurity to his soul, it is thus proper for him to do Teshuvah; See Terumos Hadeshen 2:62; Likkutei Sichos 14 p. 144 footnote 13; Sichos Kodesh Balak 5730; Shulchan Hamelech 343
 Sefer Chassidim ibid “However if he was very young when the action was committed and he does not remember, he does not need to pay”; Beis Yaakov 3; See Piskeiy Teshuvos 343:14; Lehoros Nasan 6:102
 Noda Beyehuda Tinyana Y.D. 164, brought in Pischeiy Teshuvah Y.D. 376:3