feat: NanoClaude 프로덕션 통합 — Docker, Caddy, aiosqlite 로깅

- docker-compose에 nanoclaude 서비스 추가 (포트 8100)
- Caddy /nano/* → nanoclaude 리버스 프록시 (SSE flush)
- aiosqlite 요청/응답 로깅 (request_logs 테이블)
- .env.example, CLAUDE.md 업데이트

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

nanoclaude/main.py

@@ -3,12 +3,14 @@
 from __future__ import annotations
 
 import logging
+from contextlib import asynccontextmanager
 
 from fastapi import FastAPI, Request
 from fastapi.middleware.cors import CORSMiddleware
 from fastapi.responses import JSONResponse
 
 from config import settings
+from db.database import init_db
 from routers import chat
 
 logging.basicConfig(
@@ -16,10 +18,17 @@ logging.basicConfig(
     format="%(asctime)s %(levelname)s %(name)s — %(message)s",
 )
 
+@asynccontextmanager
+async def lifespan(app: FastAPI):
+    await init_db()
+    yield
+
+
 app = FastAPI(
     title="NanoClaude",
     version="0.1.0",
     description="비동기 job 기반 AI Gateway — Phase 1",
+    lifespan=lifespan,
 )
 
 app.add_middleware(