feat(api): implement lifespan with DB and Redis connection pooling

- Refactor database.py to use init_db()/close_db() lifecycle
- Add create_db_engine() with pool_size=10, max_overflow=20, pool_pre_ping=True
- Replace cache.py stub with real Redis client using redis.asyncio
- Implement init_redis()/close_redis() with graceful error handling
- Replace no-op lifespan in main.py with proper startup/shutdown
- Enhance health endpoint to check DB and Redis connectivity
- Add tests for database, cache, and health endpoint lifecycle

Co-Authored-By: Paperclip <noreply@paperclip.ing>

api/src/cartsnitch_api/routes/health.py

@@ -1,8 +1,11 @@
 """Health check and error metrics endpoints."""
 
 from fastapi import APIRouter, Depends
+from sqlalchemy import text
 
 from cartsnitch_api.auth.dependencies import verify_service_key
+from cartsnitch_api.cache import get_redis
+from cartsnitch_api.database import get_engine
 from cartsnitch_api.middleware.error_handler import get_error_monitor
 
 router = APIRouter(tags=["health"])
@@ -10,7 +13,27 @@ router = APIRouter(tags=["health"])
 
 @router.get("/health")
 async def health():
-    return {"status": "ok"}
+    engine = get_engine()
+    db_ok = False
+    redis_ok = False
+
+    try:
+        async with engine.connect() as conn:
+            await conn.execute(text("SELECT 1"))
+        db_ok = True
+    except Exception:
+        pass
+
+    try:
+        r = get_redis()
+        if r:
+            await r.ping()
+            redis_ok = True
+    except Exception:
+        pass
+
+    status = "ok" if db_ok else "degraded"
+    return {"status": status, "db": db_ok, "redis": redis_ok}
 
 
 @router.get("/internal/error-stats", dependencies=[Depends(verify_service_key)])