Adds Vitest and Playwright testing setup with sample tests

Introduces a unified testing setup using Vitest for unit tests
and Playwright for E2E tests. Updates dependencies, adds sample
unit and E2E tests, documents test workflow, and codifies
testing and code standards in project guidelines.

Enables fast, automated test runs and improves code reliability
through enforced standards.

vitest.setup.ts

@@ -0,0 +1,2 @@
+import "@testing-library/jest-dom";
+