Let's go in reverse order of your questions:
"And what is the number in the red box on some friendly units, that appears underneath the usual strength number? Occasional enemy units have this red box too, but with no number inside it."
The red numbers indicate fatigue. If your units are not supplied and are stressed due to movement or combat, they will suffer in how well they perform. I can't remember off the top of my head, but I think combat, movement and carrying out your orders are all adversely affected by fatigue. I'm confident it is in the manual, but I'm not sure where.
Is that what's happening? That, in effect, you're getting occasional snippets of information along the lines of: "Division X was last seen two hexes south of town Y", followed by "Correction, Division X is actually one hex north of town Y"?
Your understanding of full fog of war is how I've understood it: receiving small reports of info over the course of a turn. Remember, we can't see the messenger units running around everywhere and some of those guys don't make it to their destinations. Seeing the units move where you ordered them reflects where the army staff "thinks" their forces are going, only to be corrected when the messenger arrives saying, "Hey, we're really over here..." Also, intel on the opposing forces can be even more inaccurate, so just because you see opposing units near you, doesn't mean they are exactly in that area, (or there are more of them than you realize), so be careful!
You are also correct about the extended campaigns not working properly for the latest patch. In fact, I think the problem was introduced a couple of patches ago. Frank is busy with other projects at the moment and probably doesn't have time to try and fix this any time soon. Maybe when he has a bit more time he can address it.
I've only played the 1809 scenarios and only as the Coalition. I made an Order of Battle to help me keep all those Austrians in my head. If you plan to play the 1809 scenarios I'd be happy to share them with you. I meant to make a OoB of the French army, but never got around to it. I'll have some free time in December, so maybe I'll try to attempt that project then.
I hope this was helpful,