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/database.py

@@ -1,28 +1,60 @@
 """Database session management for the API gateway."""
 
 from collections.abc import AsyncGenerator
+from typing import TYPE_CHECKING
 
 from sqlalchemy.ext.asyncio import AsyncSession, async_sessionmaker, create_async_engine
 
 from cartsnitch_api.config import settings
 
-engine = create_async_engine(
-    settings.database_url,
-    echo=False,
-    pool_size=10,
-    max_overflow=20,
-    pool_pre_ping=True,
-    pool_recycle=3600,
-)
-async_session_factory = async_sessionmaker(engine, class_=AsyncSession, expire_on_commit=False)
+if TYPE_CHECKING:
+    from sqlalchemy.engine import Engine
+
+
+_engine: "Engine | None" = None
+async_session_factory: async_sessionmaker[AsyncSession] | None = None
+
+
+def create_db_engine():
+    return create_async_engine(
+        settings.database_url,
+        pool_size=10,
+        max_overflow=20,
+        pool_pre_ping=True,
+        pool_recycle=3600,
+        echo=False,
+    )
+
+
+async def init_db() -> None:
+    global _engine, async_session_factory
+    _engine = create_db_engine()
+    async_session_factory = async_sessionmaker(_engine, class_=AsyncSession, expire_on_commit=False)
+
+
+async def close_db() -> None:
+    global _engine, async_session_factory
+    if _engine is not None:
+        await _engine.dispose()
+        _engine = None
+    async_session_factory = None
+
+
+def get_engine():
+    return _engine
 
 
 async def get_db() -> AsyncGenerator[AsyncSession, None]:
-    """FastAPI dependency that yields an async DB session."""
+    if async_session_factory is None:
+        raise RuntimeError("Database not initialized. Call init_db() first.")
     async with async_session_factory() as session:
         yield session
 
 
-async def dispose_engine() -> None:
-    """Dispose the database engine, closing all pooled connections."""
-    await engine.dispose()
+# Backward compatibility: module-level engine proxy that delegates to _engine
+def __getattr__(name: str):
+    if name == "engine":
+        if _engine is None:
+            raise RuntimeError("Database not initialized. Call init_db() first.")
+        return _engine
+    raise AttributeError(f"module {__name__!r} has no attribute {name!r}")