  Non-perturbative vacuum polarization effects are explored for a supercritical
Dirac-Coulomb system with $Z > Z_{cr,1}$ in 2+1 D, based on the original
combination of analytical methods, computer algebra and numerical calculations,
proposed recently in Refs. [1]-[3]. Both the vacuum charge density
$\rho_{VP}(\vec{r})$ and vacuum energy $\mathcal{E}_{VP}$ are considered. Due
to a lot of details of calculation the whole work is divided into two parts I
and II. Taking account of results, obtained in the part I [4] for $\rho_{VP}$,
in the present part II the evaluation of the vacuum energy $\mathcal{E}_{VP}$
is investigated with emphasis on the renormalization and convergence of the
partial expansion for $\mathcal{E}_{VP}$. It is shown that the renormalization
via fermionic loop turns out to be the universal tool, which removes the
divergence of the theory both in the purely perturbative and essentially
non-perturbative regimes of the vacuum polarization. The main result of
calculation is that for a wide range of the system parameters in the
overcritical region $\mathcal{E}_{VP}$ turns out to be a rapidly decreasing
function $\sim - \eta_{eff}\, Z^3/R\, $ with $\eta_{eff}>0 $ and $R$ being the
size of the external Coulomb source. To the end the similarity in calculations
of $\mathcal{E}_{VP}$ in 2+1 and 3+1 D is discussed, and qualitative arguments
are presented in favor of the possibility for complete screening of the
classical electrostatic energy of the Coulomb source by the vacuum polarization
effects for $Z \gg Z_{cr,1}$ in 3+1 D.
