The integrated scheduling of charging and electric bus (EB) trips has attracted significant attention and has been extensively studied in deterministic settings. However, in real-world operations, uncertainties in travel times and energy consumption can substantially influence the feasibility and effectiveness of charging plans. Moreover, vehicle-to-grid (V2G) technology is a promising solution with the potential to enhance the balance of the energy system. However, it also introduces several challenges, such as the potential reduction in battery lifespan. To date, no research has comprehensively addressed the integrated optimization of trip scheduling and charging, taking into account both V2G and battery degradation under stochastic conditions. This study proposes a two-stage stochastic optimization model to solve this joint problem, aiming to minimize the expected total operational cost. To solve the model, the sample average approximation method is employed. Numerical experiments demonstrate that the proposed model can offer a comprehensive and optimal EB system planning solution for decision-makers.