We forecast the number of galaxy clusters that can be detected via the thermal Sunyaev-Zeldovich (tSZ) signals by future cosmic microwave background (CMB) experiments, primarily the wide area survey of the CMB-S4 experiment but also CMB-S4s smaller delensing survey and the proposed CMB-HD experiment. We predict that CMB-S4 will detect 75,000 clusters with its wide survey of $f_{rm sky}$ = 50% and 14,000 clusters with its deep survey of $f_{rm sky}$ = 3%. Of these, approximately 1350 clusters will be at $z ge 2$, a regime that is difficult to probe by optical or X-ray surveys. We assume CMB-HD will survey the same sky as the S4-Wide{}, and find that CMB-HD will detect $times3$ more overall and an order of magnitude more $z ge 2$ clusters than CMB-S4. These results include galactic and extragalactic foregrounds along with atmospheric and instrumental noise. Using CMB-cluster lensing to calibrate cluster tSZ-mass scaling relation, we combine cluster counts with primary CMB to obtain cosmological constraints for a two parameter extension of the standard model ($Lambda CDM+sum m_{ u}+w_{0}$). Besides constraining $sigma(w_{0})$ to $lesssim 1%$, we find that both surveys can enable a $sim 2.5-4.5sigma$ detection of $sum m_{ u}$, substantially strengthening CMB-only constraints. We also study the evolution of intracluster medium by modelling the cluster virialization ${rm v}(z)$ and find tight constraints from CMB-S4, with further factors of 3-4 improvement for CMB-HD.
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