  In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136},
054106 (2012)], analytical results concerning the continuum percolation of
overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$
were obtained, including lower bounds on the percolation threshold. In the
present investigation, we provide additional analytical results for certain
cluster statistics, such as the concentration of $k$-mers and related
quantities, and obtain an upper bound on the percolation threshold $\eta_c$. We
utilize the tightest lower bound obtained in the first paper to formulate an
efficient simulation method, called the {\it rescaled-particle} algorithm, to
estimate continuum percolation properties across many space dimensions with
heretofore unattained accuracy. This simulation procedure is applied to compute
the threshold $\eta_c$ and associated mean number of overlaps per particle
${\cal N}_c$ for both overlapping hyperspheres and oriented hypercubes for $ 3
\le d \le 11$. These simulations results are compared to corresponding upper
and lower bounds on these percolation properties. We find that the bounds
converge to one another as the space dimension increases, but the lower bound
provides an excellent estimate of $\eta_c$ and ${\cal N}_c$, even for
relatively low dimensions. We confirm a prediction of the first paper in this
series that low-dimensional percolation properties encode high-dimensional
information. We also show that the concentration of monomers dominate over
concentration values for higher-order clusters (dimers, trimers, etc.) as the
space dimension becomes large. Finally, we provide accurate analytical
estimates of the pair connectedness function and blocking function at their
contact values for any $d$ as a function of density.
