In this paper, we consider the use of cross-layer network coding (CLNC), caching, and device-to-device (D2D) communications to jointly optimize the delivery of a set of popular contents to a set of user devices (UDs). In the considered D2D network, a group of near-by UDs cooperate with each other and use NC to combine their cached files, so as the completion time required for delivering all requested contents to all UDs is minimized. Unlike the previous work that considers only one transmitting UD at a time, our work allows multiple UDs to transmit simultaneously given the interference among the active links is small. Such configuration brings a new trade-off among scheduling UDs to transmitting UDs, selecting the coding decisions and the transmission rate/power. Therefore, we consider the completion time minimization problem that involves scheduling multiple transmitting UDs, determining their transmission rates/powers and file combinations. The problem is shown to be intractable because it involves all future coding decisions. To tackle the problem at each transmission slot, we first design a graph called herein the D2D Rate-Aware IDNC graph where its vertices have weights that judiciously balance between the rates/powers of the transmitting UDs and the number of their scheduled UDs. Then, we propose an innovative and efficient CLNC solution that iteratively selects a set of transmitting UDs only if the interference caused by the transmissions of the newly selected UDs does not significantly impact the overall completion time. Simulation results show that the proposed solution offers significant completion time reduction compared with the existing algorithms.