but on the same not do the engines actually stop spinning? I would have thought the air naturally moving through the off engine would cause it to spin too.
Yes and no. Turbojets and turbofans as well as fixed-pitch props do free-spin in the air. However, they do so at a very low rpm, usually in single-digit percent of their rated speeds. If anything, this is more of a detriment to performance because it actually acts as a big air brake. All turboprops (as well as some of the higher performance piston props) I know are equipped with variable pitch full-feathering propellers, so they actually do come to nearly a complete stop - this helps reduce their drag and increases performance in engine-out conditions. Turbojets and turbofans do have an in-flight minimum restart rpm. This can be achieved either by flying very fast, by cross-bleeding compressed air from the compressor of the working engine, or by using an auxiliary power unit (a small turbine engine designed to start the aircraft without ground assistance and to provide power when the main engines are off or failed) to feed compressed air to the air turbine starter of the failed engine.
Regardless, irrespective if the engine's internal turbo machinery remains spinning at some small fraction of rated RPM, the hot section of the engine cools off pretty quickly, since the heat source is gone and you've got very cold air going through there (not at a very high rate, but still after a few minutes of -50C air flow, it's going to be pretty much chilled). As a further example, here you have a Boeing 747-400 APU (a >1000 shp beast) starting up and going from zero to 100% rated output power in about 30 seconds. The APU is fully automatically controlled, the crew literally just flips a knob in the cockpit and that's it (here it is, near the center of the picture).
Apparently it was the turbine of an old 737-300 with the turbofan removed so one of the mech engineers told me.
Possibly an industrial variant of the CFM56. Don't know what they're called in industrial versions, I'm only familiar with GE's and some of RR's products. Industrial conversions of aviation engines do occasionally happen.
the engine was attached to a large gearbox. Maybe that's where the warm-up requirement came from
Not sure either. Gearboxes don't really need warmup either, they just need lubrication. It's mainly large castings (as occur in piston engines) that are susceptible to heat stress. Turbine engine oil has very low viscosity (far lower than automotive engine oil), so I don't think viscosity of the oil is much of a factor either... I dunno, maybe the manufacturer just wanted you to really baby the engine.