Combinatorial optimization problems arise in logistics, scheduling, and resource allocation, yet existing approaches face a fundamental trade-off among generality, performance, and usability. We present cuGenOpt, a GPU-accelerated general-purpose metaheuristic framework that addresses all three dimensions simultaneously. At the engine level, cuGenOpt adopts a "one block evolves one solution" CUDA architecture with a unified encoding abstraction (permutation, binary, integer), a two-level adaptive operator selection mechanism, and hardware-aware resource management. At the extensibility level, a user-defined operator registration interface allows domain experts to inject problem-specific CUDA search operators. At the usability level, a JIT compilation pipeline exposes the framework as a pure-Python API, and an LLM-based modeling assistant converts natural-language problem descriptions into executable solver code. Experiments across five thematic suites on three GPU architectures (T4, V100, A800) show that cuGenOpt outperforms general MIP solvers by orders of magnitude, achieves competitive quality against specialized solvers on instances up to n=150, and attains 4.73% gap on TSP-442 within 30s. Twelve problem types spanning five encoding variants are solved to optimality. Framework-level optimizations cumulatively reduce pcb442 gap from 36% to 4.73% and boost VRPTW throughput by 75-81%. Code: https://github.com/L-yang-yang/cugenopt