Exact calculations are performed on the two-dimensional strongly interacting, unpolarized, uniform Fermi gas with a zero-range attractive interaction. We briefly describe recent advances in auxiliary-field quantum Monte Carlo techniques, which enable calculations on large enough lattices to reliably compute ground-state properties in the thermodynamic limit. A new equation of state is obtained, with a parametrization provided, which can serve as a benchmark and allow accurate comparisons with experiments. The pressure, contact parameter, and condensate fraction are determined systematically vs. the interaction strength. The momentum distribution, pairing correlation, and the structure of the condensate wave function are computed. A Rashba spin-orbit coupling is then included to examine its interplay with superfluidity and the resulting pairing and spin structures and correlations.