In old times military nobility did train from childhood for combat. Similarly it was normal for children to learn the profession of the parent by helping out from early age.
Basically the only effect is that they get the skills needed at an earlier age. This is worthwhile economically for the family as the period the child is an economic drain for the family gets shorter and the period they contribute to the economic security of the family comes sooner. The same logic applies to child soldiers and assassins. You have surplus of children and a shortage of soldiers, so you make child soldiers.
An established child soldier or assassin problem would essentially be a program to get some value from children that otherwise are seen as worthless. Alternately it could be an hereditary occupation with your assassins being required to place their own children into the program. Either way the point would be to reduce the useless childhood period and make the children useful sooner.
Skill level would depend on the quality of training just as it does with adults. Hereditary system would probably be competent, but child soldiers trained because there is a shortage of soldiers would usually get minimal training and mostly learn on the job as there would also be a shortage of good trainers.
Mental and physical effects would depend on the environment and the treatment. Child soldiers in the modern world are generally heavily traumatized as in the modern world child soldiers are used by not very nice people when the situation is extremely bad. Properly institutionalized form could be much like school except more pragmatic and interesting. There is a huge difference in working because the people who killed your family will torture you otherwise or because it helps your family be fed.