I don't know much about the Non-resident application process either. You might want to call the stage G&F office and ask them.

As far as purchasing maps goes, I'm not exactly sure if the G&F website sells them or not. I buy all of mine from local sporting goods stores or the state geological survey here in town. You may just want to call the state BLM office to ask them about purchasing the BLM maps, then pick up quad maps later after you have decided which areas you are intered in hunting. The Bighorns encompass too much area to make buying quad maps for the entire range feasable.