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>

e2e/api/auth-refresh.spec.ts

@@ -0,0 +1,24 @@
+import { test, expect } from '../fixtures';
+
+test.describe('POST /auth/refresh', () => {
+  test('refreshes tokens with valid refresh token', async ({ request, testTokens }) => {
+    const res = await request.post('/auth/refresh', {
+      data: { refreshToken: testTokens.refreshToken },
+    });
+
+    expect(res.status()).toBe(201);
+    const body = await res.json();
+    expect(body).toHaveProperty('accessToken');
+    expect(body).toHaveProperty('refreshToken');
+    // New tokens should differ from original
+    expect(body.accessToken).not.toBe(testTokens.accessToken);
+  });
+
+  test('rejects invalid refresh token', async ({ request }) => {
+    const res = await request.post('/auth/refresh', {
+      data: { refreshToken: 'invalid-refresh-token' },
+    });
+
+    expect(res.ok()).toBeFalsy();
+  });
+});