I was taught (back in the 90s) that "on" had a more "universal" connotation, if that makes any sense. It meant something more like "all of us" or "people in general." It was more like the word "one" in English, as in "one can walk on the sidewalk." That is pretty similar to "we can walk on the sidewalk," but it also could imply that "people in general can walk on the sidewalk."
I think this distinction has pretty much vanished, though. Now it seems that "on" is just the common way to say "we," mainly because it is faster (because the "on" conjugations of most verbs are shorter than the "nous" conjugations).
"Est-ce que..." literally translates to "Is it that..." Often, you can replace the whole thing by "do" when you are talking about actions.
Est-ce qu'il lit un livre? = "Is it that he is reading a book" = "Does he read a book"
Here, however, it is "Est-ce qu'on est propre" = "Is it that we are clean" = "Are we clean" which is a question about our state and not our actions, so you can't replace it with "do".
I feel like "we" should not be used here for "on"..."on" more means "one" in the sense of a general idea. If you were actually saying this in conversational French, you would either mean "are you clean" or "is someone nearby clean" (which you would indicate with a gesture), or "is someone we have recently been speaking of clean".
I couldn't confirm this, but I've heard that in informal situations, people will use "on est" instead of "nous sommes" when speaking. Literally, "on est" means "one is" or "someone is" (like using the impersonal il for it, but with people). Francophones just end up using "on est" when speaking about "us" because it's easier.
Yes, "Est-on propre?" would also work.