  We study the existence, stability, and dynamics of vortex dipole and
quadrupole configurations in the nonlinear Schr\"odinger (NLS) equation on the
surface of a torus. For this purpose we use, in addition to the full
two-dimensional NLS on the torus, a recently derived [Phys. Rev. A 101, 053606
(2021)] reduced point-vortex particle model which is shown to be in excellent
agreement with the full NLS evolution. Horizontal, vertical, and diagonal
stationary vortex dipoles are identified and continued along the torus aspect
ratio and the chemical potential of the solution. Windows of stability for
these solutions are identified. We also investigate stationary vortex
quadrupole configurations. After eliminating similar solutions induced by
invariances and symmetries, we find a total of 16 distinct configurations
ranging from horizontal and vertical aligned quadrupoles, to rectangular and
rhomboidal quadrupoles, to trapezoidal and irregular quadrupoles. The stability
for the least unstable and, potentially, stable quadrapole solutions is
monitored both at the NLS and the reduced model levels. Two quadrupole
configurations are found to be stable on small windows of the torus aspect
ratio and a handful of quadrupoles are found to be very weakly unstable for
relatively large parameter windows. Finally, we briefly study the dark soliton
stripes and their connection, through a series of bifurcation cascades, with
steady state vortex configurations.
