feat(web): add Playwright E2E test suite for critical paths (#55)

Sets up @playwright/test in apps/web with playwright.config.ts
targeting localhost:3000. Adds E2E test coverage for all critical paths:
auth (login/register/validation), chat (page load, new conversation),
projects (list, empty state), settings (4 tab switches), admin (tab
switching, role guard), and navigation (sidebar links, route transitions).

Includes auth helper, separate tsconfig.e2e.json, and allowDefaultProject
ESLint config so e2e files pass the pre-commit hook. Adds pnpm test:e2e script.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

apps/web/package.json

@@ -8,6 +8,7 @@
     "lint": "eslint src",
     "typecheck": "tsc --noEmit",
     "test": "vitest run --passWithNoTests",
+    "test:e2e": "playwright test",
     "start": "next start"
   },
   "dependencies": {
@@ -21,6 +22,7 @@
     "tailwind-merge": "^3.5.0"
   },
   "devDependencies": {
+    "@playwright/test": "^1.58.2",
     "@tailwindcss/postcss": "^4.0.0",
     "@types/node": "^22.0.0",
     "@types/react": "^19.0.0",