With an increase in aerial vehicle applications, passive deformable quadrotors are getting significant attention in the research community due to their potential to perform physical interaction tasks. Such quadrotors are capable of undergoing collisions, both planned and unplanned, which are harnessed to induce deformation and retain stability by dissipating collision energies. In this article, we utilize one such passive deforming quadrotor, XPLORER, to complete various contact-rich tasks by exploiting its compliant chassis via various impact-aware planning and control algorithms. At the core of these algorithms is a novel external wrench estimation technique developed specifically for the unique multi-linked structure of XPLORER's chassis. The external wrench information is then employed for designing interaction controllers to obtain three additional flight modes: static-wrench application, disturbance rejection and yielding to the disturbance. These modes are then incorporated into a novel online exploration scheme to enable navigation in unknown flight spaces with only tactile feedback and generate a map of the environment without requiring additional sensors. Experiments show the efficacy of this scheme to generate maps of the previously unexplored flight space with an accuracy of 96.72%. Finally, we develop a novel collision-aware trajectory planner (CATAAN) to generate minimum time maneuvers for waypoint tracking by integrating collision-induced state jumps for both elastic and inelastic cases. We experimentally validate that minimum time trajectories can be obtained with CATAAN leading to a 40.38% reduction of settling time accompanied by improved tracking performance of a root mean squared error in position within 0.5cm as compared to 3cm of conventional methods.