Our activated barrier hopping theory of segmental relaxation in deeply supercooled polymer melts is applied to compute short time properties including the glassy shear modulus, localization length and vibrational frequency. Numerical calculations for specific polymers suggest the theory simultaneously predicts a reasonable elastic modulus, localized state vibrational frequency, dynamic fragility and dynamic crossover and glass transition temperatures. The theory also provides explicit connections between short time-/length-scale properties and the slow alpha relaxation process. The extension of the theory to elevated pressures is initiated. Pressure is found to broaden the deeply supercooled regime and reduce the dynamic fragility. However, the predicted Rossler–Sokolov universal supra-Arrhenius law for the temperature dependence of the alpha relaxation time remains accurate at all pressures. A common theme is the essential role played by the ratio of the dynamic crossover temperature (ideal mode coupling critical temperature) and kinetic glass transition temperature even in the deeply supercooled regime where activated processes are dominant.