"Stopover" is the word they use. Actually they call the free hotel room STPC which I think stands for "stopover complimentary".

It is not just based on time. It is also based on the fare type and fare rules for a particular flight. As you note in the rules it is not guaranteed.

My guess is that the problem is your early AM arrival and late night departure might cause some issues for the system since the hotels may not allow such early arrivals to checkin. In essence they would be on the hook for 2 nights' hotel costs because you arrive at the tail end of a night and would be in the room past check out time requiring a second night's tariff. Now, obviously lots of hotels in the world allow their early arriving guests to check in without extra costs. But maybe the hotels in Doha are not so generous. Hence the thing about "day time hours" and "night time hours".

If you choose not to fly them again over this that is obviously your choice. The strength of QR is the C class product in the air. The ground handling is poor. But the seat is tremendously comfortable. The IFE is excellent. The food is outstanding. The FA's are very good. And typically the fares they charge for all of this great service in the air are incredibly competitive with much lesser airlines.