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/e2e-testing/spec.md

@@ -0,0 +1,35 @@
+## ADDED Requirements
+
+### Requirement: E2E Test Framework
+
+The system SHALL provide end-to-end tests using Playwright.
+
+#### Scenario: Test setup
+- GIVEN the e2e test directory
+- THEN `e2e/` SHALL contain Playwright configuration
+- AND tests SHALL run against the full application stack
+
+#### Scenario: Critical user journeys
+- GIVEN the e2e test suite
+- THEN it SHALL test:
+  - User login flow
+  - Project creation and listing
+  - SSH key generation
+
+#### Scenario: Test environment
+- GIVEN e2e tests are running
+- THEN they SHALL use a dedicated test database
+- AND tests SHALL clean up data after completion
+
+### Requirement: Test Commands
+
+The system SHALL provide Makefile targets for running different test categories.
+
+#### Scenario: Make targets
+- GIVEN the Makefile
+- THEN these targets SHALL exist:
+  - `make test-unit` - Run unit tests only
+  - `make test-integration` - Run integration tests only
+  - `make test-system` - Run system tests only
+  - `make test-e2e` - Run Playwright E2E tests
+  - `make test` - Run all backend tests