This work is motivated by the engineering challenge of suppressing vibrations in turbine blades of aero engines, which often operate under extreme thermal conditions and high-Mach aerodynamic environments that give rise to complex vibration phenomena, commonly referred to as thermally-induced and flow-induced vibrations. Using Hamilton's variational principle, the system is modeled as a rotating slender Timoshenko beam under thermal and aerodynamic loads, described by a coupled system of 2*2 hyperbolic PIDEs, parabolic PDE, and ODEs, where the nonlocal terms exist in the hyperbolic PDE domain, and where the external disturbance (heat flux) flows into one boundary of the heat PDE. For the general form of such mixed systems, we present the state-feedback control design based on the PDE backstepping method, and then design an extended state observer for the unmeasurable distributed states and external disturbances using only available boundary measurements. In the resulting output-feedback closed-loop system, the state of the uncontrolled boundary, i.e., the furthest state from the control input, is proved to be exponentially convergent to zero, and all signals are proved to be uniformly ultimately bounded. Moreover, if the external disturbance vanishes, the exponential stability of the overall system is obtained. The proposed control design is validated on an aero-engine flexible blade under extreme thermal and aerodynamic conditions.