Don't think anyone was considering themselves French or German yet. There were smaller subdivisions we forgot about (or I forgot about, I'm no historian). They just happened to be all under Charlemagne's rule.

Yes, they were. Actually the core of both countries was one and the Franks did not care much about our modern borders. Western Francia became eventually France, while Eastern Francia became Franconia (roughly, the borders were always blurry and shifted a lot over time).

Of course, the Franks came from further East and never replaced the populations they invaded, so even there and then both cultures and peoples were always mixed.

Correct. The current regional concept of "Germany" proper probably started with the Treaty of Verdun, when the lands of Francia were divided amongst three of Charlemagne's grandsons. The inheritor of Francia Orientalis was in fact also known as "Louis the German".