In this thesis we investigate variational problems involving 1-dimensional sets (e.g., curves, networks) and variational inequalities related to obstacle-type dynamics from a twofold prospective. On one side, we provide variational approximations and convex relaxations of the relevant energies and dynamics, moving mainly within the framework of Gamma-convergence and of convex analysis. On the other side, we thoroughly investigate the numerical optimization of the corresponding approximating energies, both to recover optimal 1-dimensional structures and to accurately simulate the actual dynamics.