All brown sodas are Coke. Sprite and it's equivalents were separate at least where I grew up. you ask for a coke and the person taking your order asks "what kind?" and you clarify "Pepsi" or "root beer" or "coke coke".

Yes

I've lived in many locations throughout the gray area for most of my life. I've never heard "coke" used as a generic for all sodas. It's some sort of pervasive myth. Maybe that was the case at some point in history but not in the last 47 years.

Nope. I live in the DFW metro and nobody calls it coke unless it's literally a coca cola.