Yes, he should file his Federal and State returns in order to have the state taxes refunded. TurboTax requires the federal return to be filed in order to file the state. The state will start with Federal Adjusted Gross Income, as well.
Your son would check the box that he can be claimed as a dependent by someone else and that he will be claimed as a dependent.
No, you should not report his income anywhere on your return unless he has unearned income greater than $2,200 and would be subject to "Kiddie Tax."
For more information, see: Do I need to file a federal return this year?
What is the 2020 Kiddie Tax?
