It's fairly simple. In the absence of much pressure outside the rocket (which is the case when the rocket is at a suitably high altitude), the hot gasses escaping from the rocket simply keep expanding. For a deeper look as to why this occurs, you will want to look into the kinetic theory of gases.

They glow either because they are still quite hot (very near the rocket) or because they are high enough up that they are illuminated by the sun even when the sun is below the horizon from the ground.

Fairly sure it's just sunlight reflecting off the exhaust particles

The exhaust particles spread out because they're being pushed by the exhaust gasses.

Look for "expansion" and the terms "over expanded" and "under expanded".

The Wikipedia article on Space Jellyfish is a great jumping-off point too.

Any visible thermal glow is going to generally be limited to the engine nozzle itself. The primary visible aspect is reflected sunlight.

That's exactly what he said. You just reframed the answer.

There is a similar phenomenon that occurs with clouds that form in the upper atmosphere. They're called noctilucent clouds. This is the same concept that causes the exhaust to glow when the rockets are launched. The sun is below the horizon but the clouds/exhaust is high enough where the suns rays will refract off the clouds and give it an iridescent look or at least make it appear bright in the sky.

Oh neat 😯

It's just pressure. Gasses expand until the pressure is equalized. As the rocket gets higher, the exhaust gasses expand more as the ambient pressure gets lower.