refactor(proxy): modularize handlers.rs to reduce code duplication

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).

src-tauri/src/proxy/mod.rs

@@ -5,11 +5,14 @@
 pub mod circuit_breaker;
 pub mod error;
 mod forwarder;
+pub mod handler_config;
+pub mod handler_context;
 mod handlers;
 mod health;
 pub mod provider_router;
 pub mod providers;
 pub mod response_handler;
+pub mod response_processor;
 mod router;
 pub(crate) mod server;
 pub mod session;