Graph Neural Networks (GNNs) have emerged as the state-of-the-art graph learning method. However, achieving efficient GNN inference on edge devices poses significant challenges, limiting their application in real-world edge scenarios. This is due to the high computational cost of GNNs and limited hardware resources on edge devices, which prevent GNN inference from meeting real-time and energy requirements. As an emerging paradigm, device-edge co-inference shows potential for improving inference efficiency and reducing energy consumption on edge devices. Despite its potential, research on GNN device-edge co-inference remains scarce, and our findings show that traditional model partitioning methods are ineffective for GNNs. To address this, we propose GCoDE, the first automatic framework for GNN architecture-mapping Co-design and deployment on Device-Edge hierarchies. By abstracting the device communication process into an explicit operation, GCoDE fuses the architecture and mapping scheme in a unified design space for joint optimization. Additionally, GCoDE's system performance awareness enables effective evaluation of architecture efficiency across diverse heterogeneous systems. By analyzing the energy consumption of various GNN operations, GCoDE introduces an energy prediction method that improves energy assessment accuracy and identifies energy-efficient solutions. Using a constraint-based random search strategy, GCoDE identifies the optimal solution in 1.5 hours, balancing accuracy and efficiency. Moreover, the integrated co-inference engine in GCoDE enables efficient deployment and execution of GNN co-inference. Experimental results show that GCoDE can achieve up to 44.9x speedup and 98.2% energy reduction compared to existing approaches across diverse applications and system configurations.