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>

.gitignore

@@ -26,6 +26,9 @@ Thumbs.db
 
 # testing
 coverage/
+playwright-report/
+test-results/
+blob-report/
 
 # misc
 *.log