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.

openspec/changes/test-infrastructure-improvements/specs/test-organization/spec.md

@@ -0,0 +1,36 @@
+## ADDED Requirements
+
+### Requirement: Test Directory Structure
+
+The system SHALL organize tests into unit, integration, and system directories.
+
+#### Scenario: Directory layout
+- GIVEN the backend test suite
+- THEN `apps/api/tests/` SHALL contain:
+  - `unit/` - Pure logic tests with no external dependencies
+  - `integration/` - API endpoint tests with database
+  - `system/` - Full stack tests with external services
+
+#### Scenario: Running selective tests
+- GIVEN the test suite is organized
+- WHEN running `pytest -m unit`
+- THEN only unit tests SHALL execute
+- AND WHEN running `pytest -m integration`
+- THEN only integration tests SHALL execute
+
+### Requirement: Test Markers
+
+The system SHALL provide pytest markers for each test category.
+
+#### Scenario: Marker registration
+- GIVEN pytest configuration
+- THEN `pyproject.toml` SHALL register markers:
+  - `unit` - Fast tests with no external dependencies
+  - `integration` - Tests with database and external services
+  - `system` - End-to-end tests of the full stack
+
+#### Scenario: Marker usage
+- GIVEN a test file
+- THEN unit tests SHALL be marked with `@pytest.mark.unit`
+- AND integration tests SHALL be marked with `@pytest.mark.integration`
+- AND system tests SHALL be marked with `@pytest.mark.system`