It's my personal opinion that when we see such a discongreguity that we should perhaps choose other words.
What you're suggesting is basically using the etymological fallacy as a basis for changing words. Because of linguistic drift, a majority of words would have up be changed at some point - words like "nice" which came to mean their opposite, for example.
Choice of words is an individalized thing and language allows us to be as flexible as we like within still getting the point across. If I choose to say "European" instead of "western" when contrasting with indigenous Americans that sounds fine to me. If you don't make that same choice that's fine too. We'll all be understood.
Not really. Then we'd be changing terms established for centuries, that most people understand in their two contexts (geographical and cultural), with some new words we'd have to explain every time we use them -- so making things worse.
I wouldn't know if you mean "European culture" in the sense of something unique continental European (the way Europeans have unique cultural traits different from US traits, e.g. analytical vs continental philosophy), or the shared western culture Europeans and US Americans, and Australians, etc have?
Not really, only with extra context/explanation.
It's common to divide US/European culture (even though both western), or to include Japan in the western culture (even though not in the west, and their culture is not European of origin -- west in that sense is more like "westernized").