  Compact binary mergers forming in star clusters may exhibit distinctive
features that can be used to identify them among observed gravitational-wave
(GW) sources. Such features likely depend on the host cluster structure and the
physics of massive star evolution. Here, we dissect the population of compact
binary mergers in the \textsc{Dragon-II} simulation database, a suite of 19
direct $N$-body models representing dense star clusters with up to $10^6$ stars
and $<33\%$ of stars in primordial binaries. We find a substantial population
of black hole binary (BBH) mergers, some of them involving an intermediate-mass
BH (IMBH), and a handful mergers involving a stellar BH and either a neutron
star (NS) or a white dwarf (WD). Primordial binary mergers, $\sim 30\%$ of the
whole population, dominate ejected mergers. Dynamical mergers, instead,
dominate the population of in-cluster mergers and are systematically heavier
than primordial ones. Around $20\%$ of \textsc{Dragon-II} mergers are eccentric
in the LISA band and $5\%$ in the LIGO band. We infer a mean cosmic merger rate
of $\mathcal{R}\sim 12(4.4)(1.2)$ yr$^{-1}$ Gpc$^3$ for BBHs, NS-BH, and WD-BH
binary mergers, respectively, and discuss the prospects for multimessenger
detection of WD-BH binaries with LISA. We model the rate of pair-instability
supernovae (PISNe) in star clusters and find that surveys with a limiting
magnitude $m_{\rm bol}=25$ can detect $\sim 1-15$ yr$^{-1}$ PISNe. Comparing
these estimates with future observations could help to pin down the impact of
massive star evolution on the mass spectrum of compact stellar objects in star
clusters.
