  We use a recent approach [Phys. Rev. Letters, {\bf 84}, 959 (2000)] for
including Coulomb interactions in quantum systems via a classical mapping of
the pair-distribution functions (PDFs) for a study of the 2-D electron gas. As
in the 3-D case, the ``quantum temperature'' T_q of a classical 2-D Coulomb
fluid which has the same correlation energy as the quantum fluid is determined
as a function of the density parameter r_s. Spin-dependent exchange-correlation
energies are reported. Comparisons of the spin-dependent pair-distributions and
other calculated properties with any available 2-D quantum Monte Carlo (QMC)
results show excellent agreement, strongly favouring more recent QMC data. The
interesting novel physics brought to light by this study are: (a) the
independently determined quantum-temperatures for 3-D and 2-D are found to be
approximately the same, (i.e, universal) function of the classical coupling
constant Gamma. (b) the coupling constant Gamma increases rapidly with r_s in
2-D, making it comparatively more coupled than in 3-D; the stronger coupling in
2-D requires bridge corrections to the hyper- netted-chain method which is
adequate in 3-D; (c) the Helmholtz free energy of spin-polarized and
unpolarized phases have been calculated. The existence of a spin-polarized 2-D
liquid near r_s = 30, is found to be a marginal possibility. These results
pertain to clean uniform 2-D electron systems.
