The Hagahos Ashri (perek hachovel #9 in Rosh) writes that a child who damages is responsible to compensate the nizuk for his damages when he becomes an adult. He proves this from our gemara where it says that even according to the opinion that yesomim are not responsible to pay for their mu'ad animal damaging, the nizuk would collect from the caretaker, and the caretaker would collect from the orphans when they grow up. If children are responsible to compensate for their animal causing damage that occurred when they were young, they are certainly responsible to compensate for their own damage.
Most Rishonim disagree with this pesak and it is not even mentioned in Shulchan Aruch. But, according to the opinion of the hagahos ashri, would children be responsible to do teshuva for aveiros that they committed when they were younger?
R' Moshe (Y.D. 2:10) explains that even according to the hagahos ashri, the responsibility to pay when they grow up, does not apply to other aveiros. He explains that from the fact that the responsibility to compensate from damage does not require a warning in advance, the requirement to compensate does not stem from the aveirah that was violated, rather from the fact that damage was done to another. This 'avlah' that was done to harm the money of someone else, is not connected to the aveirah at all, and applies even for a child.
The only reference to a child having to do teshuva for aveiros that were violated when he was younger is the rama (o.c. 343) who says that a child who hit his father or violated other aveiros, it is good to do teshuva and something positive for kappara when he grows up. R' Moshe explains that this itself is not a requirement, rather just a good suggestion.
Regarding the proof from our gemara where the caretaker can collect from the yesomim when they grow up - perhaps that is only because the exemption of a child is based on his inability to take care of himself and watch out from damaging others. But, in the case where he has a caretaker to keep a watch on his animal, the child would be responsible for damages, just that collection is pending until he becomes an adult.