Can you get a new driving permit if you lost your original one in Michigan?
Absolutely.. Visit the Secretary of state office.
No; a person with a learners permit is required to have a licensed driver in the car. Bad Example It is illegal to drive without a valid drivers license and the law requires that we have our drivers license on us at all times when driving on public roads. So, would likely not be a good example to start off your child's driving lessons by driving without yours.
If you mean it was destroyed or you somehow lost track of it, it should be easy to get a new one from wherever you got the first one. You might have to pay a small fee. If you lost it it because of some infraction of the law, you would have to ask the judge or read the court papers that removed your permit.