I had to look this up. The inventor never trademarked the name and it was used by a competitor almost immediately. He lost a lawsuit over linoleum becoming a generic term 14 years after it's invention. There were other brand names but linoleum became a generic term almost as soon as the product was widespread.
I heard that in the South every carbonated beverage is a Coke. However, I don't understand how this works practically. If you wanted a Sprite, would you order a Coke-Sprite?
Also Heroin, but Bayer probably is fine with letting that trademark lapse. (Also Aspirin was a Bayer trademark).
It's more like if someone asks if you want a soda, they would instead ask if you want a Coke. You would then ask what kind they have. If you want a specific type at a restaurant, you would just say the kind you wanted.
No, you'd order a sprite. It's not like people order Soda-Sprite or Pop-Sprite. It only might get confusing for coke. And even then, usually, you can infer it they mean the general idea or, specifically, cocacola.