  We study the effect of spin-orbit coupling on both the zero-temperature and
non-zero temperature behavior of a two-dimensional (2D) Fermi gas. We include a
generic combination of Rashba and Dresselhaus terms into the system
Hamiltonian, which allows us to study both the experimentally relevant
equal-Rashba-Dresselhaus (ERD) limit and the Rashba-only (RO) limit. At zero
temperature, we derive the phase diagram as a function of the two-body binding
energy and Zeeman field. In the ERD case, this phase diagram reveals several
topologically distinct uniform superfluid phases, classified according to the
nodal structure of the quasiparticle excitation energies. Furthermore, we use a
momentum dependent SU(2)-rotation to transform the system into a generalized
helicity basis, revealing that spin-orbit coupling induces a triplet pairing
component of the order parameter. At non-zero temperature, we study the
Berezinskii-Kosterlitz-Thouless (BKT) phase transition by including phase
fluctuations of the order parameter up to second order. We show that the
superfluid density becomes anisotropic due to the presence of spin-orbit
coupling (except in the RO case). This leads both to elliptic vortices and
antivortices, and to anisotropic sound velocities. The latter prove to be
sensitive to quantum phase transitions between topologically distinct phases.
We show further that at a fixed non-zero Zeeman field, the BKT critical
temperature is increased by the presence of ERD spin-orbit coupling.
Subsequently, we demonstrate that the Clogston limit becomes infinite:
$T_{\rm{BKT}}$ remains non-zero at all finite values of the Zeeman field. We
conclude by extending the quantum phase transition lines to non-zero
temperature, using the nodal structure of the quasiparticle spectrum, thus
connecting the BKT critical temperature with the zero-temperature results.
