Is there any trade between Westeros and Asshai over the Sunset Sea, or are those uncharted waters?
Over the =Sunset= Sea? No. No one has ever crossed the Sunset Sea to learn what lies on the other side.
Trading ships bound for Asshai go east through the Summer Sea and the Jade Sea, which are connected by the straights at Qarth.
Also, do the Westerosi have the concept of a round world, or do they think it's flat (maybe the maesters know this...)?
Interesting question. I suspect the maesters and more educated classes realize the world is round, but the common folk may still believe in a flat world.