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Outside of all the assumptions to make this work, the real issue is that the "dark side" doesn't cool you like you're thinking. Getting rid of heat in space is actually a hard problem to solve as a vacuum is a great insulator. Heat has to be radiated away, and that takes time and lots of surface area.
Interesting and does make sense. Where does this image of stuff freezing instantly as it "enters" space vacuum come from?
Depends on the entry I suppose. A liquid being sprayed out you'd imagine would do exactly that (assuming it's not being heated by sunlight), but that's probably because spraying means it's a lot of tiny droplets, all with very little heat energy individually and lots of surface area to lose it. A big block of something like hot metal wouldn't suddenly freeze, but cool as fast as its surfaces could radiate the internal heat.
This is going strictly as a thought experiment, the math or physics is probably different to some degree.
I think an additional effect is that, the drop in pressure causes any liquids exposed to it to vaporise, which is an exothermic process, and it's a race to see whether it boils off entirely or the inner part freezes to solid from the drop in temperature through conduction. So the immediate surface of your body would either dry out or flash freeze but the inner part take a while to solidify.
Why there is any ice in space and it doesn't just sublimate away over time I'm not sure.