This study proposes an integer linear program model for ride-sharing, electric, autonomous mobility on demand (RE-AMoD) system operations and develops a model predictive control (MPC) algorithm to optimize the decisions of ride matching, vehicle routing, rebalancing, and charging. The system ensures that electric autonomous vehicles provide transportation services for up to two customers to share a ride and that they can be charged automatically during the operating period. The RE-AMoD problem is formulated as a network flow optimization problem considering ride-sharing and charging control. The objective is to minimize the customers' waiting time while minimizing the system's energy consumption. An iterative MPC is developed to compute the optimal control policy for real-time control. The case study uses real-world data from San Francisco to validate the model performance by comparing benchmark models in an RE-AMoD simulation platform and investigating the impact of ride-sharing and smart charging strategies on system performance by comparing models with no ride-sharing and heuristic charging strategies. The results show that the smart charging policy is critical for realizing ride-sharing's full advantages in RE-AMoD systems. Allowing the sharing of trips significantly improves system performance in terms of reducing fleet sizes and energy consumption while improving the customer level of service.