You probably don't need to file a return at all. You can use TurboTax free to be sure. 

If your only income is 4,000 earned (no unearned income such as interest or dividends) you won't need to file. 

If you were required to file a State Tax return you may be required to pay school tax, but that depends on the state. 

Yes, probably.  

"Any individual or estate that receives income while a resident of a taxing school district is subject to school district income tax."

There is no age or income exception,.  Since there is also no exemption deduction for dependents, you will even owe a little tax

Reference: https://tax.ohio.gov/wps/portal/gov/tax/individual/school-district-income-tax