The complexity of modern control systems necessitates architectures that achieve high performance while ensuring robust stability, particularly for nonlinear systems. In this work, we tackle the challenge of designing output-feedback controllers to boost the performance of ℓp-stable discrete-time nonlinear systems while preserving closed-loop stability from external disturbances to input and output channels. Leveraging operator theory and neural network representations, we parametrize the achievable closed-loop maps for a given system and propose novel parametrizations of all ℓp-stabilizing controllers, unifying frameworks such as nonlinear Youla parametrization and internal model control. Contributing to a rapidly growing research line, our approach enables unconstrained optimization exclusively over stabilizing controllers and provides sufficient conditions to ensure robustness against model mismatch. Additionally, our methods reveal that stronger notions of stability can be imposed on the closed-loop maps if disturbance realizations are available after one time step. Last, our approaches are compatible with the design of nonlinear distributed controllers. Numerical experiments on cooperative robotics demonstrate the flexibility of the proposed framework, allowing cost functions to be freely designed for achieving complex behaviors while preserving stability.