  We quantize Quantum Electrodynamics in $2+1$ dimensions coupled to a
Chern-Simons (CS) term and a charged spinor field, in covariant gauges and in
the Coulomb gauge. The resulting Maxwell-Chern-Simons (MCS) theory describes
charged fermions interacting with each other and with topologically massive
propagating photons. We impose Gauss's law and the gauge conditions and
investigate their effect on the dynamics and on the statistics of $n$-particle
states. We construct charged spinor states that obey Gauss's law and the gauge
conditions, and transform the theory to representations in which these states
constitute a Fock space. We demonstrate that, in these representations, the
nonlocal interactions between charges and between charges and transverse
currents, as well as the interactions between currents and massive propagating
photons, are identical in the different gauges we analyze in this and in
earlier work. We construct the generators of the Poincar\'e group, show that
they implement the Poincar\'e algebra, and explicitly demonstrate the effect of
rotations and Lorentz boosts on the particle states. We show that the
imposition of Gauss's law does not produce any ``exotic'' fractional
statistics. In the case of the covariant gauges, this demonstration makes use
of unitary transformations that provide charged particles with the gauge fields
required by Gauss's law, but that leave the anticommutator algebra of the
spinor fields untransformed. In the Coulomb gauge, we show that the
anticommutators of the spinor fields apply to the Dirac-Bergmann constraint
surfaces, on which Gauss's law and the gauge conditions obtain. We examine MCS
theory in the large CS coupling constant limit, and compare that limiting form
with CS theory, in which the Maxwell kinetic energy term is not included in the
