The fact that Balrogs can fall does not necessarily mean that they cannot fly (or change shape). First, they obviously can't do either if they're dead; from the account Gandalf gives, it seems quite possible that the Balrog died and then, as a result, fell. The phrase "cast him down" is ambiguous, and could mean anything from "pushing him down the mountain (without killing him)" to "killing him (without pushing him down the mountain)"
Even if we assume that the blow that "cast down" the Balrog did not also kill him, and that he died as a result of the fall (which I find somewhat unlikely), that still doesn't rule out shapechanging or flying abilities. We don't know how quickly or easily Valar and Maiar are able to change shape, or what restrictions exist on the shapes they can take. Obviously, they can't take literally any shape, or Melkor could have just turned himself into, say, a boulder the size of Arda, crushing and destroying every living thing. Remember when Huan fights with Sauron? Following the fight, Sauron shapeshifts into a number of different forms, but cannot elude Huan's grasp. Why didn't he just turn himself into a flea, or some other absurdly small creature, and escape? Perhaps because he couldn't, or at least he couldn't do so easily.
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