We propose an integrated approach to jointly optimize cooperative (automated) vehicle (CAV) trajectories and traffic signals, addressing uncertainties in mixed traffic at intersections. This approach consists of a stochastic signal optimization module layered on top of a robust trajectory control module. The upper signal optimization model aims to minimize the expected total delay by optimizing the signal plan, utilizing the predicted trajectories of both human-driven vehicles (HVs) and CAVs from the lower layer. The chance-constrained optimization addresses signal phase constraints and incorporates a joint probabilistic constraint on uncertain vehicle exit times, stemming from inaccurate HV motion prediction. At the lower trajectory control level, CAV accelerations are controlled using deterministic Model Predictive Control (MPC) when following other CAVs and robust MPC when following HVs. The objectives at this level are to minimize travel time and discomfort, while adhering to vehicle kinematics, safety, and red light constraints. To solve the stochastic signal optimization problem, we employ a second-order cone programming technique. The deterministic and robust MPC problems are addressed using the Pontryagin Maximum Principle and the Tube MPC (TMPC) approach, respectively. The integrated controller performance is verified through simulations under varying market penetration rates and representative traffic demand levels. The simulation results demonstrate the scalability of the proposed controller in different traffic demand levels, its robustness in stabilizing disturbances in mixed traffic flow, and its superiority in computational efficiency and performance metrics, including travel delay, fuel consumption, and emissions.