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>

package.json

@@ -29,6 +29,10 @@
     "db:seed": "prisma db seed",
     "db:studio": "prisma studio",
     "db:reset": "prisma migrate reset",
+    "test:e2e": "playwright test",
+    "test:e2e:api": "playwright test --project=api",
+    "test:e2e:web": "playwright test --project=web",
+    "test:e2e:report": "playwright show-report",
     "prepare": "husky"
   },
   "lint-staged": {
@@ -45,6 +49,7 @@
   },
   "devDependencies": {
     "@eslint/js": "^9.39.4",
+    "@playwright/test": "^1.59.1",
     "dependency-cruiser": "^17.3.10",
     "eslint": "^9.39.4",
     "eslint-config-prettier": "^10.1.8",