As Unmanned Aerial Vehicles (UAVs) are increasingly deployed in complex, cluttered, and dynamically changing environments, conventional Euclidean control architectures based on Euler-angle parameterizations exhibit fundamental limitations. These include representation singularities, gimbal lock, and degraded control performance during aggressive maneuvers such as high-speed turns, rapid attitude changes, and near-hover transitions under actuator saturation. To address these challenges, this paper proposes a unified navigation and control framework that models the multi-rotor UAV configuration directly on the Special Euclidean group SE(3). By exploiting the intrinsic geometric structure of the configuration manifold, the proposed formulation provides a singularity-free representation of coupled translational and rotational dynamics, thereby avoiding the drawbacks associated with local attitude parameterizations and under-actuation-induced singularities. Building upon the SE(3) modeling framework, an SE(3)-consistent Model Predictive Control (MPC) scheme is developed for trajectory tracking and obstacle-aware navigation. The proposed MPC explicitly incorporates physical and operational constraints, including actuator saturation limits, vehicle dynamics, and safety margins with respect to static and dynamic obstacles. This allows the controller to generate dynamically feasible control inputs while maintaining robustness in challenging flight regimes. The overall framework is implemented and evaluated using high-fidelity simulations in the MATLAB/Simulink UAV Toolbox, enabling realistic modeling of aerodynamics, sensing, and actuation. Simulation results obtained across aggressive, high-curvature flight scenarios demonstrate that the proposed geometric MPC approach achieves improved tracking accuracy, smoother control inputs, and enhanced robustness compared to conventional cascaded position–attitude control architectures. Furthermore, the computational performance remains suitable for real-time receding-horizon execution. These results indicate that geometric manifold-based control provides a principled and effective foundation for constraint-aware autonomy and reliable sim-to-real transfer in advanced multi-rotor UAV applications.