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/tests/test_cache.py

@@ -0,0 +1,50 @@
+"""Tests for Redis/DragonflyDB caching lifecycle."""
+
+import pytest
+
+from cartsnitch_api.cache import CacheClient, close_redis, get_redis, init_redis
+
+
+@pytest.mark.asyncio
+async def test_init_redis_creates_client():
+    """Test that init_redis creates the Redis client."""
+    await init_redis()
+    try:
+        r = get_redis()
+        assert r is not None
+        await r.ping()
+    finally:
+        await close_redis()
+
+
+@pytest.mark.asyncio
+async def test_close_redis_clears_client():
+    """Test that close_redis properly closes and clears the client."""
+    await init_redis()
+    await close_redis()
+    assert get_redis() is None
+
+
+@pytest.mark.asyncio
+async def test_cache_client_get_returns_none_when_not_connected():
+    """Test that CacheClient.get returns None gracefully when Redis is down."""
+    client = CacheClient()
+    # Without init_redis, get should return None
+    result = await client.get("test-key")
+    assert result is None
+
+
+@pytest.mark.asyncio
+async def test_cache_client_set_does_not_raise_when_not_connected():
+    """Test that CacheClient.set does not raise when Redis is down."""
+    client = CacheClient()
+    # Without init_redis, set should not raise
+    await client.set("test-key", "test-value", ttl_seconds=60)
+
+
+@pytest.mark.asyncio
+async def test_cache_client_delete_does_not_raise_when_not_connected():
+    """Test that CacheClient.delete does not raise when Redis is down."""
+    client = CacheClient()
+    # Without init_redis, delete should not raise
+    await client.delete("test-key")