In this paper, we aim to improve the robustness of dynamic quadrupedal locomotion through three aspects 1) fast model predictive foothold planning, 2) LQR control for robust motion tracking and 3) adaptive feet for terrain adaptation. In our proposed planning and control framework, foothold plans are updated at 400 Hz considering the current robot state and an LQR controller generates optimal feedback gains for motion tracking. The LQR optimal gain matrix with non-zero off-diagonal elements leverages the coupling of dynamics to compensate for system underactuation, such as a quadruped robot with passive ankles. The specially designed foot with adaptive sole aims at improving the traversability of rough terrains with rocks, loose gravel and rubble by enlarging the contact surfaces with ground. Experiments on the quadruped ANYmal demonstrate the effectiveness of the proposed method for robust dynamic locomotion given external disturbances and environmental uncertainties.