feat: restructure test infrastructure with unit/integration/system separation

Test Organization:
- Create tests/unit/, tests/integration/, tests/system/ directories
- Move existing tests into appropriate categories
- Add pytest markers (@pytest.mark.unit, @pytest.mark.integration)

Shared Fixtures:
- Create conftest.py with SQLite engine (for unit tests)
- Add PostgreSQL session fixture with transaction rollback
- Add TestClient fixture for API tests

Configuration:
- Update pyproject.toml with asyncio_mode=auto
- Add test markers and default addopts
- Add aiosqlite dependency for SQLite support

E2E Testing:
- Initialize Playwright in e2e/ directory
- Add playwright.config.ts
- Create login flow E2E test

Build:
- Add test-unit, test-integration, test-system, test-e2e to Makefile
- Update test target to run all categories
- Add testing documentation to README

Note: Some tests have import issues due to missing python-jose
package in dev environment. This needs to be addressed separately.

apps/api/pyproject.toml

@@ -25,7 +25,15 @@ dev = [
     "mypy>=1.7.0",
     "ruff>=0.1.0",
     "httpx>=0.25.0",
+    "aiosqlite>=0.19.0",
 ]
 
 [tool.pytest.ini_options]
 pythonpath = ["."]
+asyncio_mode = "auto"
+markers = [
+    "unit: Fast tests with no external dependencies",
+    "integration: Tests with database and external services",
+    "system: End-to-end tests of the full stack",
+]
+addopts = "-m 'not system'"