feat(e2e): add Playwright E2E testing infrastructure and critical path tests

Set up Playwright with dual-project config (API + Web), auth test fixtures,
16 E2E tests covering registration, login, profile, token refresh, and
homepage rendering. Added GitHub Actions CI workflow for automated E2E runs.

Co-Authored-By: Paperclip <noreply@paperclip.ing>

e2e/fixtures/index.ts

@@ -0,0 +1,3 @@
+export { test, expect } from './auth.fixture';
+export { createTestUser, registerUser, loginUser } from './auth.fixture';
+export type { TokenPair } from './auth.fixture';