Is all digital sensor noise thermal?
And what is the relationship between temperature and noise? For example, as we cooled a CMOS towards absolute zero, would we discover that high-ISO images would be less noisy? Would they converge towards a truly noiseless image at maximum ISO and absolute zero?
No. The noise is in the photon count, and it is the square root of the number of photons. So if you get 9 photons, the noise level is 3 (33%); if you get 10000 photons, the noise level is 100 (1%).
Photons are not perfectly distributed over space, but randomly, and even at perfect 0 K, different receiver buckets will catch different amounts of photons each time you try.
There is dark current, as inkista wrote in a comment, and there's quantization noise at the digitizer, and there's gain noise in the analog amplifiers, and there's readout noise ( slightly less than 100% of collected charge is extracted).
When people write "scene-limited" or "photon -limited" noise, what they mean is that the photon shot noise is much greater than the combination of all the electronics noise sources. You can never do better than being photon-limited. Luckily, if you read up a bit on energy per photon, you'll see that an SNR of sqrt(number of photons) is a very large number -- unless you're imaging distant galaxies or some such :-)