I'm not sure, hence the question. AFAIK temperature only comes into play at inference time once the distribution is known, but I don't know if there are other places where random numbers are involved.

Yes, lots of other randomness.

Eg you tend to randomly shuffle your corpus to train on. If you use drop-out (https://en.wikipedia.org/wiki/Dilution_(neural_networks)) you use randomness. You might also randomly perturb your training data. Lots of other sources of randomness that you might want to try.