  We perform large-scale, numerically exact calculations on the two-dimensional
interacting Fermi gas with a contact attraction. Reaching much larger lattice
sizes and lower temperatures than previously possible, we determine
systematically the finite-temperature phase diagram of the
Berezinskii-Kosterlitz-Thouless (BKT) transitions for interaction strengths
ranging from BCS to crossover to BEC regimes. The evolutions of the pairing
wavefunctions and the fermion and Cooper pair momentum distributions with
temperature are accurately characterized. In the crossover regime, we find that
the contact has a non-monotonic temperature dependence, first increasing as
temperature is lowered, and then showing a slight decline below the BKT
transition temperature to approach the ground-state value from above.
