Can schizophrenia occur in a young child?

Yes, but rarely does. For reasons still poorly understood, schizophrenia is relatively rare until after puberty, but there are well-described cases in children as young as 4 years old. Aftter puberty, the risk rises, with the peak risk at 18-20. Research suggests this is related to major changes in brain architecture that occurs during puberty, but we still do not understand what causes the disorder.
Yes. Yes but not necessarily in the form seen in adults. Autistic spectrum disoders can be confused with schizophrenia.