I'm pretty sure the answer is yes. I've used it from A to B (or was it B to A) and I don't think there's anything to stop you going back and forth!
Qatar airlines seem to have started using gates C (which are a bit more crowded in my experience) and there seem to be 2 skytrains above. I don't understand the layout of the airport sufficiently to know if they're the same trains as between A and B, but I think they're different. You could explore and map the Doha skytrain system I suppose!
If you're going on to Bangkok there's now a satellite terminal building connected to the main building by an underground train. That's more strictly controlled so I don't think you could go back and forth at Suvarnabhumi.
Hope you enjoy Doha!