Traditional rigid robots offer high load capacity and precise motion control, but struggle with irregular object contact and safe human interaction, capabilities that soft robots achieve through material compliance. However, the design of soft robotic systems still relies heavily on intuition and bio-inspired heuristics. Due to the strong coupling between geometric nonlinearity and contact mechanics, intuition- and heuristic-based approaches fail to fully exploit the design space, highlighting the need for systematic computational frameworks. Topology optimization provides this rigor, with the level-set method being particularly attractive for generating smooth, manufacturable boundaries. While level-set topology optimization has been widely applied to linear problems, its extension to nonlinear hyperelastic materials, particularly for compliant mechanisms and soft robots, remains largely unexplored. This paper presents a unified topology optimization framework for neo-Hookean hyperelastic materials, with shape sensitivities rigorously derived using the adjoint method and including higher-order nonlinear terms. The framework is validated using benchmark problems, including a mean-compliance test and a compliant gripper, demonstrating accuracy and robustness. In addition, a boundary-capturing and extraction algorithm is developed to explicitly track evolving pneumatic chambers during level-set propagation, enabling the consistent application of design-dependent pressure loads on moving boundaries. Optimized designs, including displacement inverters and both pneumatic and non-pneumatic compliant grippers, are fabricated and experimentally tested, showing strong agreement with numerical predictions. These results establish a comprehensive computational framework for nonlinear level-set topology optimization with hyperelastic materials and enable the systematic design of soft robotic structures and compliant mechanisms with embedded actuation.