  We consider experimental constraints in the MeV region in order to determine
the parameter space for the $U(1)_X$ extension of the Standard Model, presented
in the first part of our work. In particular, we focus on the model
UV-completed by cold WIMPs. We conclude that the electron anomalous magnetic
moment, the neutrino trident production and the relic abundance $\Omega_{CDM}$
provide the most stringent bounds and, in particular cases, they are sufficient
to exclude dark-photon ($A'$) models. By allowing the axial-vector coupling of
the dark gauge boson $Z'$, the interference effect with the SM gauge bosons may
reduce the bounds coming from the trident neutrino production. At the same
time, such coupling allows a region of the parameter space already favored both
by the relic abundance and by the discrepancy between experimental result and
theoretical prediction for the muon anomalous magnetic moment. We emphasize
that light-$Z'$ interactions, non-universal for the two first lepton families,
necessarily create a difference in the proton charge radius measured in the
Lamb shift of the $e$-hydrogen and $\mu$-hydrogen. Finally, we determine the
effects of the new gauge boson on the forward-backward asymmetry in $e^+ e^-
\rightarrow \bar{f} f$, $f = \mu, \tau$, and on the leptonic decays $M
\rightarrow j \nu_j l^+ l^-$, where $M = \pi, K, D, D_s, B$ and $j,l = e, \mu$.
