We observed a remarkable improvement of domain wall (DW) mobility, DW velocity, giant magnetoimpedance (GMI) effect and magnetic softening at appropriate stress-annealing conditions. Beneficial effect of stress-annealing on GMI effect and DW dynamics is associated with the induced transverse magnetic anisotropy. An improvement of the circumferential permeability in the nearly surface area of metallic nucleus is evidenced from observed magnetic softening and remarkable GMI effect rising. We assumed that the outer domain shell with transverse magnetic anisotropy associated to stress-annealing induced transverse magnetic anisotropy affects the travelling DW in a similar way as application of transversal bias magnetic field allowing enhancement the DW velocity. Observed decreasing of the half-width of the EMF peak in stress-annealed microwires can be associated to the decreasing of the characteristic DW width. Consequently, stress annealing enabled us to design the magnetic anisotropy distribution beneficial for optimization of either GMI effect or DW dynamics.