We show how nanomechanical strain can be used to dynamically control the optical response of self-assembled quantum dots embedded in nanomechanical bridges, giving a tool to shift electron and hole levels, manipulate mechanoexciton shape, orientation, fine-structure splitting, and optical transitions, transfer carriers between dots, and interact qubits for quantum processing. Conversely, we show how modulation of the quantum dot optical response can be used to monitor locally an applied nanomechanical strain. Atomistic tight-binding theory is used to describe the response of electrons and holes in a self-assembled quantum dot to applied nanomechanical strain. The internal strain due to the lattice mismatch, the nanomechanical strain, and the internal atomic readjustment to minimize the applied strain must all be accounted for to model correctly the strain effects. Electrons and hole levels and charge distributions can shift together or in opposite directions depending on how the strain is applied. This gives control for tailoring band gaps and optical response. The strain can also be used to transfer electrons and holes between vertically or laterally coupled dots, giving a mechanism for manipulating transition strengths and interacting qubits for quantum information processing. Applied strain can be used to manipulate the fine-structure splitting of mechanoexcitons by distorting electron and hole charge distributions and rotating hole orientation. Most importantly, nanomechanical strain reengineers both the magnitude and phase of the exciton exchange coupling to tune exchange splittings, change the phase of spin mixing, and rotate the polarization of mechanoexcitons, providing phase and energy control of excitons.