Personally having transitted DOH a few times I would really enjoy to have 12 daylight hours in DOH to explore. The city is fairly compact and quite interesting. The museum of Islamic Art has a great reputation and time spent walking around the Souk Waquif in my opinion would sure beat any of the facilities on offer at the airport itself. I have unfortunately only ever had 8 hour night transfers and while the souk is alive and vibrant at that time the museum is closed and you cannot really see much of the city.
My only experience of the airport itself consists of a few duty free shops and time in the lounge.