We investigate the combined effect of a far - infrared cavity field and Rashba spin - orbit interaction on the band structure and transport properties of artificial graphene composed of quasi-2D InAs/GaAs quantum dots. The coupling to cavity photons is modeled by constructing a complete basis as the tensor product of the electronic Hilbert space and the Fock space. Our numerical calculations for the system embedded in a linear cavity predict the existence of both type-I and type-II Dirac points which can be distinguished by their response to Rashba interaction. Namely, Rashba coupling opens a gap at type-II Dirac points, while type-I Dirac points remain gapless. The possibility of gap-opening for type-II Dirac points is demonstrated analytically as well. For both cylindrical and linear cavities, we demonstrate the formation of electron - photon hybrid states and Rabi splittings between energy minibands. Multiple crossings, and anticrossings between Dirac-band replicas produce pronounced modifications of the spin-Hall conductivity, including strong anisotropy and oscillatory behavior controlled by cavity geometry and polarization of photons. Our results show that the interplay between Rashba and cavity couplings governs Dirac-point physics and provides a route toward tunable polaritonic spin-transport in engineered nanostructures.