We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter mu(a) > 0. We establish observability inequalities for weakly (when mu(a) is an element of [0, 1[) as well as strongly (when mu(a) is an element of [1, 2[) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e., when mu(a) > 2) and prove the blowup of the observability time when mu(a) converges to 2 from below. Thus, using the Hilbert uniqueness method we deduce the exact controllability of the corresponding degenerate control problem when mu(a) is an element of [0, 2[. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary-damped wave equation for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim., 51 (2005), pp. 61-105], [F. Alabau-Boussouira, J. Differential Equations, 249 (2010), pp. 1473-1517], together with our results for linear damping, to perform this stability analysis.