Collaborative perception significantly enhances individual vehicle perception performance through the exchange of sensory information among agents. However, real-world deployment faces challenges due to bandwidth constraints and inevitable calibration errors during information exchange. To address these issues, we propose mmCooper, a novel multi-agent, multi-stage, communication-efficient, and collaboration-robust cooperative perception framework. Our framework leverages a multi-stage collaboration strategy that dynamically and adaptively balances intermediate- and late-stage information to share among agents, enhancing perceptual performance while maintaining communication efficiency. To support robust collaboration despite potential misalignments and calibration errors, our framework prevents misleading low-confidence sensing information from transmission and refines the received detection results from collaborators to improve accuracy. The extensive evaluation results on both real-world and simulated datasets demonstrate the effectiveness of the mmCooper framework and its components.