This paper presents a high-order discontinuous Galerkin (DG) method for shallow water equations with bottom topography that simultaneously preserves the well-balanced property, positivity of water depth, and physically admissible bounds on flow velocity. This is achieved through a novel estimate grounded in Riemann invariants. Developed within a hydrostatic reconstruction framework, the scheme discretizes flux gradients and source terms in a compatible manner, ensuring the exact preservation of the "lake-at-rest" steady state over non-flat bathymetry. To suppress spurious oscillations near discontinuities while maintaining high-order accuracy, we introduce an entropy-induced Convex Oscillation Suppressing (COS) limiting strategy. Formulated as a convex correction, this limiter retains high resolution in smooth regions and is meticulously designed to be compatible with the well-balanced discretization, ensuring that equilibrium states remain unperturbed after limiting. A distinctive feature of the proposed method is the rigorous treatment of fluid velocity in near-dry regions. Rather than resorting to conventional desingularization techniques or ad hoc velocity cut-offs when water depth vanishes, we develop a novel Riemann-invariant-bound-preserving approach backed by rigorous theoretical analysis. By enforcing invariant-region constraints at the discrete level, this mechanism prevents numerical blow-up of the velocity u = (hu)/h as depth approaches zero, providing a unified framework for controlling both depth positivity and velocity boundedness. Under a suitable CFL condition, the method theoretically guarantees cell-average positivity and well-balancedness. Extensive numerical experiments in one and two dimensions-encompassing non-flat topography and dynamic wet/dry interfaces-confirm the high-order accuracy, robustness, and structure-preserving properties of the proposed scheme.