Edge inference (EI) has emerged as a promising paradigm to address the growing limitations of cloud-based Deep Neural Network (DNN) inference services, such as high response latency, limited scalability, and severe data privacy exposure. However, deploying DNN models on resource-constrained edge devices introduces additional challenges, including limited computation/storage resources, dynamic service demands, and heightened privacy risks. To tackle these issues, this paper presents a novel privacy-aware optimization framework that jointly addresses DNN model deployment, user-server association, and model partitioning, with the goal of minimizing long-term average inference delay under resource and privacy constraints. The problem is formulated as a complex, NP-hard stochastic optimization. To efficiently handle system dynamics and computational complexity, we employ a Lyapunov-based approach to transform the long-term objective into tractable per-slot decisions. Furthermore, we introduce a coalition formation game to enable adaptive user-server association and design a greedy algorithm for model deployment within each coalition. Extensive simulations demonstrate that the proposed algorithm significantly reduces inference delay and consistently satisfies privacy constraints, outperforming state-of-the-art baselines across diverse scenarios.