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/main.py

@@ -26,10 +26,14 @@ from cartsnitch_api.routes.user import router as user_router
 
 @asynccontextmanager
 async def lifespan(app: FastAPI):
-    await cache_client.initialize()
+    from cartsnitch_api.database import init_db, close_db
+    from cartsnitch_api.cache import init_redis, close_redis
+
+    await init_db()
+    await init_redis()
     yield
-    await cache_client.close()
-    await dispose_engine()
+    await close_redis()
+    await close_db()
 
 
 def create_app() -> FastAPI: