Out-of-state students attending a Michigan college or university are considered nonresidents. You may be required to file a nonresident state return.

If your federal adjusted gross income is over $4,050 then you are required to file a nonresident Michigan state return. However, the filing requirement is different if you were claimed as a dependent. In that case, you are required to file if your federal adjusted gross income is over $1,500 (single or married filing separately) or over $3,000 if filing a married filing joint return. 

Even if you are not required to file you may want to in order to claim a refund of tax withheld. Follow the directions in the TurboTax FAQ below if you need to prepare a nonresident state return. 

https://ttlc.intuit.com/replies/3302052

No, you do not need to file a MI return. Full time students, who use their parents (or their own) Ohio residence address are covered by the reciprocity agreement.

Ohio has a reciprocal agreement, on wages,  with all neighboring states. No state taxes are withheld or due and you do not normally need to file a return for the other state (MI, PA, KY, WV, IN). But OH will tax you on your other state income.