Add Chess quality gates and managed Playwright smoke

[sarthakagrawal927]

Summary

• add Biome correctness/suspicious linting and repair the small existing findings needed for a clean gate
• make TypeScript cover the app, API, config, and Playwright test
• add a managed Vite Playwright configuration so the smoke test uses port 5173 consistently
• run lint, typecheck, build, and the browser smoke in CI

Dependency review

• adds @biomejs/biome 2.5.5 as a development-only Fleet-standard lint tool
• no production/runtime dependency change
• code-cleanup guard recorded the direct devDependency and lockfile delta

Validation

• pnpm lint
• pnpm typecheck
• pnpm build
• pnpm test (1 passed)
• ./scripts/docs-validate.sh
• git diff --check

Production deployment, Vercel status, and docs-domain decisions remain operator-owned and are tracked with the blocked follow-up.

Closes #7