It's a complicated association. Some will say that mouth breathing will cause a narrowed maxilla, others will say that a narrow maxilla will cause mouth breathing. I myself have a well-developed maxilla and don't have a narrow arch. Nevertheless I have some nasal congestion issues that are related to chronic sinusitis. I'm not sure why I can breathe better through my nose now, it's still a mystery to me. I've always been a mouth breather at night, though I usually don't have daytime issues breathing through my nose. Most pictures have me with my mouth closed.
In regards to the upturned nose issue, I think this is mostly genetic but can signal vertical maxillary overgrowth. Others can correct me if I'm wrong. I actually have a degree of maxillary prognathism, and a mild mandibular recession, that together give me a more obvious class II jaw appearance.