  We investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition in a
two-dimensional (2D) Fermi gas with spin-orbit coupling (SOC), as a function of
the two-body binding energy and a perpendicular Zeeman field. By including a
generic form of the SOC, as a function of Rashba and Dresselhaus terms, we
study the evolution between the experimentally relevant equal
Rashba-Dresselhaus (ERD) case and the Rashba-only (RO) case. We show that in
the ERD case, at fixed non-zero Zeeman field, the BKT transition temperature
$T_{BKT}$ is increased by the effect of SOC for all values of the binding
energy. We also find a significant increase in the value of the Clogston limit
compared to the case without SOC. Furthermore, we demonstrate that the
superfluid density tensor becomes anisotropic (except in the RO case), leading
to an anisotropic phase-fluctuation action that describes elliptic vortices and
antivortices, which become circular in the RO limit. This deformation
constitutes an important experimental signature for superfluidity in a 2D Fermi
gas with ERD SOC. Finally, we show that the anisotropic sound velocities
exhibit anomalies at low temperatures, in the vicinity of quantum phase
transitions between topologically distinct uniform superfluid phases.
