Can brown shades on a vinyl rug bring out the charm of small spaces?

That would be a good idea. Almost any color rug will go well with white walls. Dark brown can create a dramatic contrast between the walls and the floor, but will make the room feel smaller, perfect if you're going for cozy. Light brown will create a warmer and welcoming atmosphere. If you want to go even lighter, cream rugs will make rooms feeling airy and light, while adding warmth.