I will be flying out of town for a week, can I park at the hotel for seven nights without an additional charge?

I did leave my car there once when I had to fly out for a few days. They let me do it but they weren't too excited about it

I doubt it, especially in the winter they need to plow the snow. There are several hotels near the DTW airport that you can park & stay.

Your best bet is to check with the front desk just to be sure.