If someone has a bench warrant for outstanding DUI case, would he or she be able to get a license in another state?

That depends on the State. Many will check and catch the warrant. The person should go and take care of the matter.

The other state would be very likely to learn about the warrant. You have a unique driver's id number which follows you from state to state. If you have a warrant for failure to appear, you might be arrested at any time. If this arrest takes place in another state you might get a very uncomfortable ride back to Michigan and spend a lot of time in custody before ever seeing the judge. People dragged in from other states do not do well on being granted bond. I recommend you get an attorney and turn yourself in to face the charges. People who walk in on their own have a better chance of walking out. The best time is a morning, early in the week, not a week with holidays. The best place to turn yourself in is at the courthouse where you failed to appear. If you cannot afford a lawyer you can do the turn-in on your own. Good luck.

Usually there are questions on the application asking if you have been licensed in another state and the answers are usually done under penalty of perjury.

The DMV won't catch the warrant, because they couldn't care less about it. What they will catch is the fact that this person's license is suspended, and that is what will prevent this person from getting a license in any state.

Not if his license is suspended because of the DUI

Probably not. But assuming a misdemeanor DUI, a CA lawyer can clear the bench warrant for you without you appearing in person.

Generally not, her record will be.