The structural, electronic, and vibrational properties of glassy GexSe1−x are studied using density-functional-based molecular dynamics. The focus is on four compositions (x=10%,20%,25%,33%) spanning the rigidity transitions and representing typical compositions of flexible, intermediate, and stressed rigid systems. We investigate structural properties including structure factors, pair distribution functions, angular distributions, coordination numbers, and neighbor distributions and compare our results with experimental findings, when available. Most noticeable is the excellent agreement found in the reproduction of the structure in real and reciprocal space which allows tracking the effect of Ge composition on the structure. Ring statistics and ring correlations are examined and followed across the rigidity transition, and the details of typical small rings show a much more complex picture than established previously. A comparison is made with simple bond models and their validity is discussed. Topological constraint analysis is performed and shows that the onset of rigidity changes substantially the angular motion inside the Ge tetrahedra, which displays increased soft bending motions in the stressed rigid phase. We then investigate the vibrational properties via the vibrational density of states and the dielectric function (infrared absorption), and discuss them with respect to experimental findings. Finally, the electronic properties are computed and show an excellent agreement with respect to previous first-principles simulations and to experiments.