  In this paper, we are concerned with solutions to the following nonlinear
Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u
+ \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\,
\R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$
where $N \geq 1$, $\mu =\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$
and the parameter $\lambda \in \R$ appearing as Lagrange multiplier is unknown.
In the mass subcritical case, we establish the compactness of any minimizing
sequence to the minimization problem given by the underlying energy functional
restricted on the constraint. As a consequence of the compactness of any
minimizing sequence, orbital stability of minimizers is derived. In the mass
critical and supercritical cases, we investigate the existence, radial symmetry
and orbital instability of solutions. Meanwhile, we consider the existence,
radial symmetry and algebraical decay of ground states to the corresponding
zero mass equation with defocusing perturbation. In addition, dynamical
behaviors of solutions to the Cauchy problem for the associated dispersive
equation are discussed.
