Resource allocation is investigated to enhance the performance of device-to-device (D2D) cooperation in a fog radio access network (F-RAN) architecture. Our envisioned framework enables two D2D links to share certain orthogonal radio resource blocks (RRBs) by forming device-clusters. In each device-cluster, both content-holder device-users (DUs) transmit to the content-requester DUs via an enhanced remote radio head (eRRH) over the same RRBs. Such RRBs are shared with the uplink F-RAN as well. The intra device-cluster interference is mitigated by exploiting both uplink and downlink rate-splitting schemes, and the inter device-cluster interference is mitigated by using an orthogonal RRB allocation strategy. Our objective is to maximize the end-to-end sum-rate of the device-clusters while reducing the interference between D2D cooperation and the uplink F-RAN over the shared RRBs. Towards this objective, a joint optimization of device-clustering, transmit power allocations, assignment of device-clusters to the eRRHs, and allocation of RRBs among the device-clusters is presented. Since the joint optimization is NP-hard and intractable, it is decomposed into device-clustering and resource allocation sub-problems, and efficient solutions to both sub-problems are developed. Based on the solutions to the sub-problems, a semi-distributed and convergent algorithm, entitled rate-splitting for multi-hop D2D (RSMD), is proposed to obtain the device-clusters and resource allocation for these device-clusters. Through extensive simulations, efficiency of the proposed RSMD algorithm over several benchmark schemes is demonstrated