Motivated by recent experiments on atomic Dirac fermions in a tunable honeycomb optical lattice, we study the attractive Hubbard model of superfluidity in the anisotropic honeycomb lattice. At weak-coupling, we find that the maximum mean field pairing transition temperature, as a function of density and interaction strength, occurs for the case with isotropic hopping amplitudes. In this isotropic case, we go beyond mean field theory and study collective fluctuations, treating both pairing and density fluctuations for interaction strengths ranging from weak to strong coupling. We find evidence for a sharp sound mode, together with a well-defined Leggett mode over a wide region of the phase diagram. We also calculate the superfluid order parameter and collective modes in the presence of nonzero superfluid flow. The flow-induced softening of these collective modes leads to dynamical instabilities involving stripe-like density modulations as well as a Leggett-mode instability associated with the natural sublattice symmetry breaking charge-ordered state on the honeycomb lattice. The latter provides a non-trivial test for the experimental realization of the one-band Hubbard model. We delineate regimes of the phase diagram where the critical current is limited by depairing or by such collective instabilities, and discuss experimental implications of our results.
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