Monocular simultaneous localization and mapping (SLAM) algorithms estimate drone poses and build a 3D map using a single camera. Current algorithms include sparse methods that lack detailed geometry, while learning-driven approaches produce dense maps but are computationally intensive. Monocular SLAM also faces scale ambiguities, which affect its accuracy. To address these challenges, we propose an edge-aware lightweight monocular SLAM system combining sparse keypoint-based pose estimation with dense edge reconstruction. Our method employs deep learning-based depth prediction and edge detection, followed by optimization to refine keypoints and edges for geometric consistency, without relying on global loop closure or heavy neural computations. We fuse inertial data with vision by using an extended Kalman filter to resolve scale ambiguity and improve accuracy. The system operates in real time on low-power platforms, as demonstrated on a DJI Tello drone with a monocular camera and inertial sensors. In addition, we demonstrate robust autonomous navigation and obstacle avoidance in indoor corridors and on the TUM RGBD dataset. Our approach offers an effective, practical solution to real-time mapping and navigation in resource-constrained environments.