  We study relativistic mean-field models with hadron masses and coupling
constants depending self-consistently on a scalar meson field. We demonstrate
that by the field redefinition some models can be equivalently transformed into
each other. Thereby the large variety of scaling functions for masses and
couplings can be reduced to a restricted set of functions with a constrained
dependence on a scalar field. We show how by choosing properly the latter
scaling functions one may stiffen or soften the equation of state at high
densities and simultaneously increase the threshold density for the direct Urca
process without any change of the description of nuclear matter close to the
saturation density. The stiffening of the equation of state might be motivated
by recent neutron star mass measurements, whereas the increase of the threshold
density for the direct Urca process is required by the analysis of neutron star
cooling data. We demonstrate that if a rho meson is included in a mean-field
model as a non-Abelian gauge boson, then there is a possibility for a charged
rho-meson condensation in dense nuclear matter. We show that such a novel phase
can be realized in neutron star interiors already for sufficiently low
densities, typically $\sim 3\div 4 n_0$, where $n_0$ is the nuclear saturation
density. In the framework of the relativistic mean field model the new phase
arises in a second-order phase transition. The appearance of a $\rho^-$
condensate significantly alters the proton fraction in a neutron star but
changes moderately the equation of state. The neutrino emissivity of the
processes involving a $\rho^-$ meson condensate is estimated.
