  Effects of neutrino charge radius and magnetic moment constraints obtained
from the astrophysical observations and reactor experiments, as well as
in-medium modifications of the weak and electromagnetic nucleon form factors of
the matter on the neutrino electroweak interaction with dense matter, are
estimated. We use a relativistic mean-field model for the in-medium effective
nucleon mass and quark-meson coupling model for nucleon form factors. We
analyze the neutrino scattering cross section and mean free path in cold
nuclear matter. We find that the increase of the cross section relative to that
without neutrino form factors results in the decrease of the neutrino mean free
path when neutrino form factors and the in-medium modifications of the nucleon
weak and electromagnetic form factors are simultaneously considered. The
quenching of the neutrino mean free path is evaluated to be about 12-58\% for
the values of $\mu_\nu = 3 \times 10^{-12} \mu_B$ and $R_\nu = 3.5 \times
10^{-5}~\textrm{MeV}^{-1}$ compared with that obtained for the $\mu_\nu =0$ and
$R_\nu =0$. The decrease of the neutrino mean free path is expected to
decelerate the cooling of neutron stars. Each contribution of the neutrino form
factors to the neutrino mean free path is discussed.
