That stage 2 engine nozzle really glows hot! Looks hot enough to melt, but holding together impressively. Also, the exhaust into the vacuum looks like a messy grey cloud, not the bright cone I might have expected. Very cool. Or hot.
Yes! Amazingly, although the nozzle looks just like a bell-shaped piece of sheet metal, it in fact uses "regenerative cooling" which means the nozzle is full of small pipes which wind back and forth like a radiator. The rocket fuel is passed through these pipes before combustion, resulting in a heat exchange which is doubly beneficial, keeping the nozzle (relatively) cool and also heating up the rocket fuel so it combusts more efficiently.
> the exhaust into the vacuum looks like a messy grey cloud, not the bright cone I might have expected
Yep, since there is no atmospheric pressure squeezing it into a cone shape, the exhaust just kind of flies everywhere!
Merlin 1C Vacuum had a regeneratively cooled combustion chamber but radiatively cooled expansion nozzle. I'm not sure if Merlin 1D Vacuum differs from this though.
You're correct for the last generation of engines - Merlin 1C Vacuum was radiatively cooled. However I haven't found any confirmation one way or the other for the 1D Vacuum engine, which they're using currently. It's possible they haven't made this info public yet due to ITAR.