Multi-Agent collaboration addresses inherent limitations of individual agent systems, including limited sensing range and occlusion-induced blind spots. Despite significant progress have been achieved, persistent challenges such as constrained communication bandwidth and under-explored subsequent extensions still hinder real-time deployment and further developments of collaborative autonomous driving systems. In this work, we propose ZeRCP, a unified communication-efficient framework that bridges collaborative perception with future scene prediction. Specifically, (i) we devise a plug-and-play request-free spatial filtering module (ZeroR) that eliminates the reliance on request maps while preserving inter-agent spatial complementarity modeling. This approach further reduce communication latency and bandwidth consumptions. (ii) We design a multi-scale pyramidal prediction network anchored by a novel Spatial-Temporal Deformable Attention (STDA) module, extending frame-wise detection to multi-frame predictions. This method adeptly models spatiotemporal dynamics without relying on auto-regressive recursion. We evaluate our method on a large-scale dataset in challenging semantic segmentation and scene prediction tasks. Extensive experiments demonstrate the superiority and effectiveness of ZeRCP in bandwidth-constrained collaboration scenarios and spatiotemporal prediction applications.