I think the main reason Lar deSouza(and others) draws robes like long coats is that they look like dresses otherwise.
However, I prefer to draw them like a monk's robes, right over the head. Possibly with jeans and a tee-shirt underneath. The only exception is Sirius, and you know how he dresses. (Bare chest, ripped robes, and ragged pants.)
I don't think there would be any school crest on the robes themselves, and I don't recall anything special being mentioned in the various school clothing-lists, but there might be decoration on the cloak clasps.