Previously well-established experimental trends of ground-state properties of crystalline and amorphous Mo1-xSix and Mo1-xNix alloys with 0 ≤ x ≤ 1 are predicted by first-principles calculations and inter-relationship with sound velocities and elastic properties are identified. The free energy of mixing, calculated at 300 K provides accurate values of the critical Si and Ni concentrations leading to the crystalline-to-amorphous transition. Specifically, a transition from BCC to amorphous state is predicted for xSi ~ 0.14 and xNi ~ 0.32, while FCC to amorphous transition is observed for xNi ~ 0.77. These structural transitions are accompanied by modifications of the out-of-plane longitudinal and shear elastic constants. In the crystalline region, a pronounced softening of elastic moduli is predicted for both solid solution alloys. While for amorphous Mo-Ni, the elastic constants do not undergo significant changes, for amorphous Mo-Si, they exhibit two distinct behaviors depending on the electronic properties and bonding character. For 0.2 ≤ xSi ≤ 0.7, the metallic character of the amorphous alloys is maintained and the elastic properties are remarkably stable. For xSi > 0.7, a progressive increase in the atomic volume is observed and the amorphous alloys acquire a covalent character and a reduced coordination number. Surprisingly, during this transition, both the longitudinal and shear acoustic velocities continuously increase, despite a progressive softening of the elastic stiffness constants. These calculations provide deep insight at the atomic-scale and reproduce satisfactorily the earlier experimental works of magnetron co-sputtered Mo-Si films, while some disagreement on elastic properties remain for more energetic ion beam sputtered Mo-Ni films, likely partially attributed to point-defects.