I only speculated about the Van Allen belts. ISS is in LEO, and our planet looks very large--larger than the guest planets appear on TOS, so I assume it is between geosynch and LEO. Looks about--what, 1,000? or 2,500--to--5,000 miles up or so? Beyond the van allen belts?
It doesn't matter how big the planet "looks" unless you know how big the planet is and what its rotational characteristics are. If you have a planet rotating fast enough (or with weak enough gravity), you could be in geostationary orbit at an altitude of fifteen feet. The Van Alan Belts, on the other hand (not all planets will even have them) go between 1000 and 5000 km, with a little break, then a much larger belt at 25,000 km and extending pretty much forever (tapering off slowly into space).
I was just eyeballing the distance over an Earth sized--of course, the effects folks actually made the Enterprise look as if it were going "around" the planet by turning the model. In truth the ship would almost look as if it were going straight up close.
Strictly speaking, "up close" the Enterprise would look as if it were stationary and the PLANET was slowly revolving underneath it.
Addition - the only things that changed are the color and rotation direction of the planet. Sometimes it went with the Enterprise's direction of travel and sometimes it did not. I haven't watched enough of them yet to say for sure if the FX guys tried to match the dialogue regarding "orbits", but it does seem like there was some effort beyond the re-use of the Enterprise motion
I would need to see an example of that, since as far as I know this is not the case. The closest thing I've heard to this is the trick where the generic "planet" element is used with Enterprise orbiting from a front aspect, then is used at a different time--flipped horizontally--with Enterprise orbiting from a rear aspect, in which case the planet rotation appears to be the opposite direction.
Those shots were a composite of the background starfield, the rotating planet, and the ship doing its orbiting path.
Here are a few that I recently remembered seeing that you can see the different rotation FX used in: The Mark of Gideon -> planet spinning counterclockwise (if viewing axis from the top of the screen down) Enemy Within -> planet spinning clockwise (axis tilted toward viewer and to the left) Miri -> planet spinning counterclockwise (axis tilted to the right and similar to Enemy Within since it is probably a reverse angle as you described) Court Martial -> planet spinning clockwise (axis tilted to the left)
Since the planet(globe) was repainted every time, it could have been spun clockwise or counter clockwise during the shooting.