I’ve been running a 10sp chainset with my 9sp transmission for ages. Works fine.

Yes 10spd rings on 9spd fine, 9spd rings on 10spd may be an issue.
9spd is slightly wider than 10 spd, so you may have issues with the narrow chain on a wider ring. It’ll probably be fine though.

After hearing lots of conflicting opinions on this I spent an hour or so with a set of Shimano 9sp cranks and 10sp cranks and a vernier calliper accurate to 0.02mm.

I could find absolutely no difference in width, tooth height, profile, spacing on the cranks, between 9sp and 10sp chainrings. So 10sp chainrings will play happily in an otherwise 9sp system, and vice versa.

They do, however come in different ratios. Whereas the standard 9sp rings gave you a nice wide 44/32/22 spacing, the ten speed ones are 42/32/24 as they expect you to have a wider ratio 11/36 cassette on the back to make up the difference.

And by the way, the answer to the next question is that 10sp chains are the same internal width as 9sp, but slightly narrower external width – so yes, the plates are thinner/flatter.