  The observed flavor-changing neutral-current (FCNC) processes in the standard
model (SM) arise from the loop diagrams involving the weak charged currents
mediated by the $W$-gauge boson. Nevertheless, the top-quark FCNCs and
lepton-flavor violating processes resulting from the same mechanism are highly
suppressed. We investigate possible new physics effects that can enhance the
suppressed FCNC processes, such as $t\to q(h,V)$ with $V=\gamma,Z,g$, $h\to
\ell \ell'$, and $\ell\to \ell' \gamma$. To achieve the assumption that the
induced-FCNCs are all from quantum loops, we consider the scotogenic mechanism,
where a $Z_2$ symmetry is introduced and only new particles carry an odd $Z_2$
parity. With the extension of the SM to include an inert Higgs doublet, an
inert charged Higgs singlet, a vector-like singlet quark, and two neutral
leptons, it is found that, with relevant constraints taken into account, the
$t\to c (h, Z)$, $h\to \mu\tau$, and $\tau\to \ell \gamma$ decays can be
enhanced up to the expected sensitivities in experiments. The branching ratios
of $h\to \mu^+ \mu^-/\tau^+ \tau^-$ from only new physics effects can reach up
to ${\cal O}(10^{-3})$. Intriguingly, the resulting muon $g-2$ can fit the
combined data within $2\sigma$ errors, whereas the electron $g-2$ can have
either sign with a magnitude of ${\cal O}(10^{-13}-10^{-12})$. In addition, we
examine the oblique parameters in the model and find that the resulting
$W$-mass anomaly observed by CDF II can be accommodated.
