A new disturbance observer based control scheme is developed for a quadrotor under the concurrent disturbances from a lightweight elastic tether cable and a lumped vertical disturbance. This elastic tether is unusual as it creates a disturbance proportional to the multicopter's translational movement. This paper takes an observer-based approach to estimate the stiffness coefficient of the cable and uses the system model to update the estimates of the external forces, which are then compensated in the control action. Given that the tethered cable force affects both horizontal channels of the quadrotor and is also coupled with the vertical channel, the proposed disturbance observer is constructed to exploit the redundant measurements across all three channels to jointly estimate the cable stiffness and the vertical disturbance. A pseudo-inverse method is used to determine the observer gain functions, such that the estimation of the two quantities is decoupled and stable. Compared to standard disturbance observers which assume nearly constant disturbances, the proposed approach can quickly adjust its total force estimate as the tethered quadrotor changes its position or tautness of the tether. This is applied to two experiments - a tracking performance test where the multicopter moves under a constant tether strain, and an object extraction test. In the second test, the multicopter manipulates a nonlinear mechanism mimicking the extraction of a wedged object. In both cases, the proposed approach shows significant improvement over standard Disturbance Observer and Extended State Observer approaches. A video summary of the experiments can be found at https://youtu.be/9gKr13WTj-k.