He doesn't ever turn upside down in real life, unlike Walter's goldfish, because he is able to compensate for the slosh with corresponding movements of his pectoral fins. It's actually pretty neat to watch him keep himself fairly stationary when his water is in motion. I think that's mostly what his speck of a brain does, calculate the amount and direction of propulsion necessary to either a) remain more or less in one place, b) drift around gently, or c) surface for a bite of air or a pellet.
I think Bela does do something that approximates sleeping, because sometimes he gets very still and doesn't seem to notice me waving my hands at him because I'm afraid he might be dead. Usually I can spy some slight correcting motion in his fins, which I suppose is like the motions humans make in response to their environments without actually waking up -- scratching or coughing or covering their faces. But sometimes, especially when he is cozied up in the motionless water inside one of his jars, even his fins are still and I am compelled to rap on the glass to put my mind at rest -- whereupon he darts away as if he had been stung. Sorry, Bela.
It seems unlikely that he dreams, because he has no frontal lobes. But then I'm not sure that the frontal lobes are what one dreams with. Do fish dream, Doctor Three?