Agile quadrotor flight in cluttered scenes requires more than a reactive mapping from a depth image to a control command: the vehicle must remember which regions have been observed, infer nearby occupied space, and act under partial visibility and tight latency. In this paper, we present Mapping-Aware Dreamer (MAD), a geometry-aware world model for vision-based quadrotor flight. Instead of using raw-image reconstruction as the main self-supervised objective, MAD learns recurrent latent dynamics that reconstruct robocentric occupancy and visibility grid maps together with proprioceptive states. This design forces the latent state to encode local geometry, visibility history, and ego-motion in a form that is directly relevant to collision avoidance. MAD is trained in DiffAero using a GPU-parallel map-construction module that provides high-throughput supervision for occupancy and visibility. The learned representation is used in three policy-learning modes: imagination-based MAD-Dreamer and feature-extractor variants based on PPO and SHAC. Across visual navigation and racing tasks, MAD-based agents achieve higher success rates, faster flight, and better cross-task transfer than corresponding vision-only baselines. The model also produces interpretable map predictions and accurate ego-motion estimates from depth observations. We further deploy the learned policy on a physical quadrotor with an Intel RealSense D435i and demonstrate safe indoor and outdoor flight under limited sensing, reaching 9.66 m/s in simulation and 5.05 m/s in real-world forest experiments. These results show that mapping-aware world models provide a practical middle ground between modular aerial navigation and end-to-end learning.