I think if you don't go all the way across the 36" it will be wasted space. I think you could still achieve the stand alone look with it going across. It could be open below, or it could have legs, or look like a free standing chest of drawers. But I would want it to go across because the small areas to either side would be another place to clean that may be hard to get to. If the sides are open enough for easier cleaning then you might not have enough counter space. And a gal needs a place to put her purse if she's visiting:) Good luck, it sounds like a fun little project.

I would also use the full width, for the reasons enduring stated. You could change the look by leaving out the sidesplashes, or even omit the standard backsplash too. I've seen lovely powder rooms where the one wall above the sink was tiled to the ceiling with decorative tile (like glass mosaic). That could be a great option to give it some style.