It is not necessary for a 3-year-old to write, and a child that young is not being naughty if he is not interested. As for the school behavior, look up "social anxiety" or "social anxiety in children" for information.
He is only 3 year old I think it will get better ,he may not be too interested in writing at this age ,they do like to read and have stories told to them.What is the school like does the teacher stimulate the children ? When he is home do some writing and reading, children love stories .it may help feed the interest. Dont be too concerned yet , he is very young .good Luck