This paper studies the interplay between device-to-device (D2D) communications and real-time monitoring systems in a cellular-based Internet of Things (IoT) network. In particular, besides the possibility that the IoT devices communicate directly with each other in a D2D fashion, we consider that they frequently send time-sensitive information/status updates (about some underlying physical processes) to their nearest base stations (BSs). By modeling the locations of the IoT devices as a bipolar Poisson Point Process (PPP) and that of the BSs as another independent PPP, we characterize the performance of the D2D links and status update links in terms of network throughput and Age-of-Information (AoI), respectively. We consider a maximum power constraint and distance-dependent fractional power control for all status update transmissions. Hence, the locations of the IoT devices allowed to send status updates are constrained to lie within the Johnson-Mehl cells. For this set-up, the average network throughput is obtained by deriving the mean success probability of the D2D links, whereas the spatial moments of the temporal mean AoI are obtained by deriving the moments of the temporal means of both success and scheduling probabilities of the status update links.