Hello and welcome to MDC.
I'm sorry you're having a hard time finding a midwife. It sounds like Ohio is not the most friendly state for midwifery, so there may be some underground. What has your search looked like so far? Maybe we can help you with brainstorming with how to find a midwife. If one wants a midwife, one should ideally have a midwife.
To answer your questions: to get a birth certificate, call your local vital statistics office and ask what you need to do to register a birth that has not been attended by a medical professional. The requirements are different in different states (but may not be a big deal at all. Maybe proof of pregnancy, maybe an affidavit or two from someone saying the child is yours...)
I've only had one doctor ask where my child was born (in the interest of making conversation,) so you won't necessarily have to deal with it. You could say you don't feel comfortable divulging the midwife's name (implying that you had one,) which would be reasonable in a state where midwifery is not legally fully supported.