This paper develops an innovative approach to the modeling and analysis of downlink cellular networks with device-to-device (D$2$D) transmissions. The analytical embodiment of the signal-to-noise and-interference ratio (SINR) analysis in general fading channels is unified due to the H-transform theory, a taxonomy never considered before in stochastic geometry-based cellular network modeling and analysis... The proposed framework has the potential, due to versatility of the Fox's H functions, of significantly simplifying the cumbersome analysis procedure and representation of D$2$D and cellular coverage, while subsuming those previously derived for all the known simple and composite fading models. By harnessing its tractability, the developed statistical machinery is employed to launch an investigation into the optimal design of coexisting D$2$D and cellular communications. We propose novel coverage-aware power control combined with opportunistic access control to maximize the area spectral efficiency (ASE) of D$2$D communications. Simulation results substantiate performance gains achieved by the proposed optimization framework in terms of cellular communication coverage probability, average D$2$D transmit power, and the ASE of D$2$D communications under different fading models and link- and network-level dynamics. (read more)