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_database.py

@@ -0,0 +1,62 @@
+"""Tests for database initialization and lifecycle."""
+
+import pytest
+from sqlalchemy.ext.asyncio import AsyncSession, async_sessionmaker, create_async_engine
+
+from cartsnitch_api.database import (
+    close_db,
+    create_db_engine,
+    get_engine,
+    init_db,
+)
+
+
+@pytest.mark.asyncio
+async def test_create_db_engine_creates_engine_with_pool_settings():
+    """Test that create_db_engine creates engine with correct pool settings."""
+    engine = create_db_engine()
+    assert engine is not None
+    pool = engine.pool
+    assert pool.size() == 10
+    assert pool._max_overflow == 20
+    await engine.dispose()
+
+
+@pytest.mark.asyncio
+async def test_init_db_sets_engine_and_factory():
+    """Test that init_db properly initializes the engine and session factory."""
+    await init_db()
+    try:
+        eng = get_engine()
+        assert eng is not None
+        from cartsnitch_api import database
+
+        assert database.async_session_factory is not None
+    finally:
+        await close_db()
+
+
+@pytest.mark.asyncio
+async def test_close_db_disposes_engine():
+    """Test that close_db properly disposes the engine."""
+    await init_db()
+    await close_db()
+    assert get_engine() is None
+    from cartsnitch_api import database
+
+    assert database.async_session_factory is None
+
+
+@pytest.mark.asyncio
+async def test_get_db_yields_session_after_init():
+    """Test that get_db yields working sessions after init_db."""
+    await init_db()
+    try:
+        from cartsnitch_api.database import get_db
+
+        gen = get_db()
+        session = await gen.__anext__()
+        assert isinstance(session, AsyncSession)
+        await gen.aclose()
+    finally:
+        await close_db()