For internal photon states, is it necessary to sum over the longitudinal polarization state in addition to the transverse states? And if so, does the ordinary Feynman-gauge propagator take care of this?
No. However, the longitudinal states do not couple to the conserved electromagnetic current, so it doesn't matter whether you include them or not. Thus the ##k^\mu k^\nu## term in the propagator can be multiplied by an arbitrary number (this is sometimes called "generalized Feynman gauge"), and the scattering amplitudes do not depend on that number.