  We present a theoretical study of the superfluidity and the corresponding
collective modes in two-component atomic Fermi gases with s-wave attraction and
synthetic Rashba spin-orbit coupling. The general effective action for the
collective modes is derived from the functional path integral formalism. By
tuning the spin-orbit coupling from weak to strong, the system undergoes a
crossover from an ordinary BCS/BEC superfluid to a Bose-Einstein condensate of
rashbons. We show that the properties of the superfluid density and the
Anderson-Bogoliubov mode manifest this crossover. At large spin-orbit coupling,
the superfluid density and the sound velocity become independent of the
strength of the s-wave attraction. The two-body interaction among the rashbons
is also determined. When a Zeeman field is turned on, the system undergoes
quantum phase transitions to some exotic superfluid phases which are
topologically nontrivial. For the two-dimensional system, the nonanalyticities
of the thermodynamic functions and the sound velocity across the phase
transition are related to the bulk gapless fermionic excitation which causes
infrared singularities. The superfluid density and the sound velocity behave
nonmonotonically: they are suppressed by the Zeeman field in the normal
superfluid phase, but get enhanced in the topological superfluid phase. The
three-dimensional system is also studied.
