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_routes/test_health.py

@@ -0,0 +1,77 @@
+"""Tests for health check endpoint."""
+
+import pytest
+from unittest.mock import AsyncMock, patch
+
+from cartsnitch_api.database import init_db, close_db
+
+
+@pytest.mark.asyncio
+async def test_health_returns_db_and_redis_fields(client):
+    """Test that health endpoint returns db and redis status fields."""
+    from cartsnitch_api.cache import init_redis, close_redis
+
+    await init_db()
+    await init_redis()
+
+    try:
+        response = await client.get("/health")
+        assert response.status_code == 200
+        data = response.json()
+        assert "status" in data
+        assert "db" in data
+        assert "redis" in data
+    finally:
+        await close_redis()
+        await close_db()
+
+
+@pytest.mark.asyncio
+async def test_health_returns_degraded_when_db_down():
+    """Test that health returns degraded when database is down."""
+    from cartsnitch_api.database import _engine
+    from cartsnitch_api.routes.health import health
+
+    # Simulate engine is None (DB not initialized)
+    with patch("cartsnitch_api.routes.health.get_engine", return_value=None):
+        response = await health()
+        assert response["status"] == "degraded"
+        assert response["db"] is False
+
+
+@pytest.mark.asyncio
+async def test_health_returns_ok_when_db_up(client):
+    """Test that health returns ok when database is up."""
+    from cartsnitch_api.database import init_db, close_db
+    from cartsnitch_api.cache import init_redis, close_redis
+
+    await init_db()
+    await init_redis()
+
+    try:
+        response = await client.get("/health")
+        assert response.status_code == 200
+        data = response.json()
+        if data["db"]:
+            assert data["status"] == "ok"
+    finally:
+        await close_redis()
+        await close_db()
+
+
+@pytest.mark.asyncio
+async def test_health_redis_down_does_not_make_unhealthy(client):
+    """Test that Redis being down does not make health return unhealthy."""
+    from cartsnitch_api.database import init_db, close_db
+
+    await init_db()
+
+    try:
+        response = await client.get("/health")
+        data = response.json()
+        # Redis being down should not make status "degraded"
+        # Only DB failure makes it degraded
+        if not data["db"]:
+            assert data["status"] == "degraded"
+    finally:
+        await close_db()