Providing direct communications among a rapidly growing number of wireless devices within the coverage area of a cellular system is an attractive way of exploiting the proximity among them to enhance coverage and spectral and energy efficiency. However, such device-to-device (D2D) communications create a new type of interference in cellular systems, calling for rigorous system analysis and design to both protect mobile users (MUs) and guarantee the connectivity of devices. Motivated by the potential advantages of cognitive radio (CR) technology in detecting and exploiting underutilized spectrum, we investigate CR-assisted D2D communications in a cellular network as a viable solution for D2D communications, in which devices access the network with mixed overlay-underlay spectrum sharing. Our comprehensive analysis reveals several engineering insights useful to system design. We first derive bounds of pivotal performance metrics. For a given collision probability constraint, as the prime spectrum-sharing criterion, we also derive the maximum allowable density of devices. This captures the density of MUs and that of active macro base stations. Limited in spatial density, devices may not have connectivity among them. Nevertheless, it is shown that for the derived maximum allowable density, one should judiciously push a portion of devices into receiving mode in order to preserve the connectivity and to keep the isolation probability low. Furthermore, upper bounds on the cellular coverage probability are obtained incorporating load-based power allocation for both path-loss and fading-based cell association mechanisms, which are fairly accurate and consistent with our in-depth simulation results. Finally, implementation issues are discussed.