  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-S4's 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
($\Lambda CDM+\sum m_{\nu}+w_{0}$). Besides constraining $\sigma(w_{0})$ to
$\lesssim 1\%$, we find that both surveys can enable a $\sim 2.5-4.5\sigma$
detection of $\sum m_{\nu}$, 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. The binned cluster counts, Fisher
matrices, and other associated products can be downloaded from
https://github.com/sriniraghunathan/tSZ_cluster_forecasts.

Cosmology and Nongalactic Astrophysics