  In this paper, we are concerned with the standing waves for the following
nonlinear Schr\"{o}dinger equation $$i\partial_{t}\psi=-\Delta
\psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast
|\psi|^2)\psi-
  \lambda_3|\psi|^p \psi,~~~ (t,x)\in \mathbb{R}^+\times \mathbb{R}^3,$$ where
$0<p<4$. We mainly study the existence and stability/instability properties of
standing waves for this equation, in two cases: the first one is that no
magnetic potential is involved, (i.e. $b=0$ in the equation) and the second one
is that $b\neq 0$. To be precise, in the first case, by considering a
minimization problem on a suitable Pohozaev manifold we prove the existence of
ground states, and show further that all ground state standing waves are
strongly unstable by blow-up in finite time. Moreover, by making use of the
ideas of their proofs, we are able to prove the existence and instability of
normalized solutions, whose proofs seem to be new, compared with the studies of
normalized solutions in the existing literature. In the second case, the
situation is more difficult to be treated, due to the additional term of the
partial harmonic potential. We manage to prove the existence of stable standing
waves for $p\in (0,4)$ and with some assumptions on the coefficients, where
solutions are obtained as global minimizers if $p\in (0,\frac{4}{3}]$, and as
local minimizers if $p\in [\frac{4}{3}, 4)$. In the mass-critical and
supercritical cases $p\in [\frac{4}{3}, 4)$, we establish the variational
characterization of the ground states on a suitable manifold which is different
from the one neither of the Nehari type nor of the Pohozaev type, and then
prove the existence of ground states. Finally under some assumptions on
$\omega$ and $p$, we prove that the ground state standing waves are strongly
unstable.
