Cmdstan provides warmups. Can I use those as part of samples used to estimate posterior or is sampling and warmup fundamentally different?
Fundamentally different. The warmup realizations are not a valid Markov Chain across the different windows and are biased at the beginning.
I am asking this question because in my case I see good mixing in warmups (and no divergencies) starting from 10th draw. Instead of using warmup of 500 I can I set it to 50 and expect that sampling will behave as with warmup of 500. Just to shave time…
It is not even doing any adaptation within the first 50 draws. So, what you are observing is in no way general. If it happens to be true in a particular case, then you could reduce the warmup but it probably runs very fast even at the defaults.