In this paper, we consider a cooperative transmission model for video applications and services (VASs) in dense device-to-device (D2D) networks. The model enables the mobile users (MUs) to flexibly receive the videos from macro base station (MBS) and D2D networks with mobile edge caching. Particularly, we formulate a multi-rate selection and power allocation assisted probabilistic edge caching (MPC) optimisation problem under the resource constraints on storage, bandwidth, and power. This problem is solved for the optimal caching probabilities of requested videos corresponding to proper encoding rates selected. The optimal powers of caching MUs and MBS for transmitting the videos are also found to maximise the playback quality, while utilising the system resources. The MPC optimisation problem, which is complicated due to the presence of binary and real variables and various constraints, is feasibly solved by genetic algorithms (GA) with penalty function and truncated chromosome. Simulation results are shown to demonstrate the benefits of both GA and MPC methods compared to other benchmarks. Detailed analyses and interesting findings provide useful insights into the mobile edge caching design of dense D2D networks for VASs.