Electrification and distributed energy resources (DERs) are vital for reducing the building sector's carbon footprint. However, conventional reactive control is insufficient in addressing many current building-operation-related challenges, impeding building decarbonization. To reduce building carbon emissions, it is essential to consider dynamic grid electricity mix and incorporate the coordination between DERs and building energy systems in building control. This study develops a novel model predictive control (MPC)-based integrated energy management framework for buildings with multiple DERs considering dynamic grid electricity mix and pricing. A linear, integrated high-fidelity model encompassing adaptive thermal comfort, building thermodynamics, humidity, space conditioning, water heating, renewable energy, electric energy storage, and electric vehicle, is developed. An MPC controller is developed based on this model. To demonstrate the applicability, the developed framework is applied to a single-family home with an energy management system through whole-year simulations considering three climate zones: warm, mixed, and cold. In the simulations, the framework reduces the whole-building electricity costs and carbon emissions by 11.9% - 38.3% and 7.2% - 25.1%, respectively, compared to conventional control. Furthermore, the framework can reduce percent discomfort time from 25.7% - 47.4% to nearly 0%, compared to conventional control. The framework also can shift 86.4% - 100% of peak loads to off-peak periods, while conventional control cannot achieve such performance. The case study results also suggest that pursuing cost savings is possible in tandem with carbon emission reduction to achieve co-benefits (e.g., simultaneous 37.7% and 21.9% reductions in electricity costs and carbon emissions, respectively) with the proposed framework.