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/cache.py

@@ -1,9 +1,41 @@
 """Redis/DragonflyDB caching helpers."""
 
+import logging
+from typing import TYPE_CHECKING
+
 import redis.asyncio as redis
+from redis.asyncio import Redis
 
 from cartsnitch_api.config import settings
 
+if TYPE_CHECKING:
+    from cartsnitch_api.config import Settings
+
+logger = logging.getLogger(__name__)
+
+_redis: "Redis | None" = None
+
+
+def get_settings() -> "Settings":
+    return settings
+
+
+async def init_redis() -> None:
+    global _redis
+    _redis = redis.from_url(settings.redis_url)
+    await _redis.ping()
+
+
+async def close_redis() -> None:
+    global _redis
+    if _redis is not None:
+        await _redis.aclose()
+        _redis = None
+
+
+def get_redis() -> Redis | None:
+    return _redis
+
 
 class CacheClient:
     """Redis/DragonflyDB caching with connection pooling.