You have to do the tear-down inspection if the prop hits the runway....regardless of whether the engine is running, windmilling, or stopped. At least cut the mixture, you'll still have to the tear-down but you probably won't have to replace anything.
Not necessarily. I actually had this happen to me in a PA-31. I pulled the mixtures and props all together. The props stopped in the right position, but because they were 3 blade, they still got a little nicked up. My company called Lycoming and were informed that if the props were within limits and did not require overhaul, the engines did not need a tear down. The plane was flying a week later with filed props, a new radome, new nose gear doors, and a few antennae. (And a new hydraulic line for the one that failed in the first place.)