  We use N-body simulations to study the dynamical evolution of Population III
(Pop III) stellar systems and the resulting binary statistics. We design a
physically-motivated framework for the initial conditions of Pop III star
clusters, based on small-scale hydrodynamic simulations and the scale-free
nature of disk evolution during Pop III star formation. Our novel approach
enables us to explore the dependence of binary statistics on initial conditions
and arrive at more robust predictions for the signals of Pop III X-ray binaries
(XRBs) and binary black hole (BBH) mergers, compared to simple extrapolations
of Pop III protostar systems. We find that binary properties are highly
sensitive to the initial cluster size and distribution of binary separation,
while the effect of initial mass function is relatively minor. Our simulations
predict less close binaries, and thus, significantly lower efficiencies (by a
factor of $\sim 10-10^{4}$) for the formation and accretion of Pop III XRBs,
than found in previous studies, implying that the contribution of Pop III XRBs
to the cosmic X-ray background is negligible and their feedback effects are
unimportant. We estimate the efficiency of Pop III BBH mergers as $\sim
10^{-5}-10^{-4}\ \rm M_{\odot}^{-1}$, for which 3-body hardening by surrounding
stars in dense star clusters or close binary interactions is required to
facilitate in-spirals of BBHs. All simulation data, including catalogs of Pop
III binaries and multiple systems, are publicly available.
