Deep-learning-based intelligent services have become prevalent in cyber-physical applications, including smart cities and health-care. Deploying deep-learning-based intelligence near the end-user enhances privacy protection, responsiveness, and reliability. Resource-constrained end-devices must be carefully managed to meet the latency and energy requirements of computationally intensive deep learning services. Collaborative end-edge-cloud computing for deep learning provides a range of performance and efficiency that can address application requirements through computation offloading. The decision to offload computation is a communication-computation co-optimization problem that varies with both system parameters (e.g., network condition) and workload characteristics (e.g., inputs). However, deep learning model optimization provides another source of tradeoff between latency and model accuracy. An end-to-end decision-making solution that considers such computation-communication problem is required to synergistically find the optimal offloading policy and model for deep learning services. To this end, we propose a reinforcement-learning-based computation offloading solution that learns optimal offloading policy considering deep learning model selection techniques to minimize response time while providing sufficient accuracy. We demonstrate the effectiveness of our solution for edge devices in an end-edge-cloud system and evaluate with a real-setup implementation using multiple AWS and ARM core configurations. Our solution provides 35% speedup in the average response time compared to the state-of-the-art with less than 0.9% accuracy reduction, demonstrating the promise of our online learning framework for orchestrating DL inference in end-edge-cloud systems.