Extract common request handling logic into dedicated modules:
- handler_config.rs: Usage parser configurations for each API type
- handler_context.rs: Request lifecycle context management
- response_processor.rs: Unified streaming/non-streaming response handling
Reduces handlers.rs from ~1130 lines to ~418 lines (-63%), eliminating
repeated initialization and response processing patterns across the
four API handlers (Claude, Codex Chat, Codex Responses, Gemini).
