Should a disconnected CPU fan spin freely if I spin it with the finger

BIOS resetting is a certainty. And since you didn't do it manually(choosing to reset the BIOS in the BIOS!), you should replace the CMOS battery with a new one.

That doesn't explain CPU temp being high.

Enable Q-fan in the BIOS. Q-fan may just be an ASUS thing, but you have an ASUS board.

Then at least the fans should quieten down even before the going into windows stage. It'll be interesting if they then speed up again and how soon.

You do need to address the CPU overheating.

You could open the computer and check that the CPU fan is spinning fast.. And check the temp immediately on booting windows. And check the temp in the BIOS if the BIOS shows CPU temp..

And if you're not worried about static possibly damaging a part(I'd risk it on my comp for sure, and on somebody elses with a warning to them so getting an ok first!), then feel the heatsink and see if it is very hot. It could be misreporting the temp. Maybe there are other ways of viewing or measuring the temp of the cores.

I've opened my case, took the cooler off and cleaned it a bit (was full of dust/dirt). I've left the thermal grease there because I don't have any replacement right now.

There's your problem. Whenever you take the cooler off the CPU, you need to clean off the remnants of the thermal paste, using strong (90%+) isopropyl alcohol, and reapply new paste. Never leave the old paste there, because air bubbles will form within it, reducing heat transfer between the CPU die and the heatsink. That's why your processor is overheating.

I strongly recommend you not run the computer until you've replaced the thermal paste. If you cannot wait until the AS5 arrives, get some no-name paste from any computer store - even the cheapest paste you can find, as long as it's applied properly, is better than reusing old paste, even if it's high-end stuff.