We introduce a robust numerical strategy for the numerical simulation of several free-boundary problems arising in the study of nonlinear wave-structure interactions in shallow-water flows. We investigate two types of boundary-evolution equations: (i) a kinematic-type equation, associated with the interaction of waves with a moving lateral wall, (ii) a fully-nonlinear singular equation modeling the evolution of the interface between a solid obstacle placed on the surface and the fluid. At the continuous level, the flow is globally modeled with the hyperbolic Nonlinear Shallow-Water (NSW) equations, including varying topography, and at the discrete level, an arbitrary-order discontinuous Galerkin (DG) method is stabilized with a Local Subcell Correction (LSC) method. Mimicking the theoretical study of the continuous problem, suitable diffeomorphisms are introduced to recast the moving-boundary problems into fixed-boundary ones, and to compute the boundary's evolution through an Arbitrary-Lagrangian-Eulerian (ALE) description. For any order of polynomial approximation, the resulting global algorithm is shown to: (i) preserve the Discrete Geometric Conservation Law (DGCL), (ii) ensure the preservation of the water height positivity at the sub-cell level, (iii) preserve the class of motionless steady-states (well-balancing), possibly with the occurrence of a partly immersed obstacle. Several numerical computations highlight that the proposed strategy: (i) effectively approximate the new free-boundary IBVPs introduced in [19], (ii) is able to accurately handle strong flow singularities without any robustness issues, (iii) retains the highly accurate subcell resolution of DG schemes.