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-profile.spec.ts

@@ -0,0 +1,29 @@
+import { test, expect } from '../fixtures';
+
+test.describe('GET /auth/profile', () => {
+  test('returns user profile for authenticated user', async ({ authedRequest, testUser }) => {
+    const res = await authedRequest.get('/auth/profile');
+
+    expect(res.status()).toBe(200);
+    const body = await res.json();
+    expect(body).toHaveProperty('id');
+    expect(body.phone).toBe(testUser.phone);
+    expect(body.fullName).toBe(testUser.fullName);
+  });
+
+  test('rejects unauthenticated requests', async ({ request }) => {
+    const res = await request.get('/auth/profile');
+
+    expect(res.ok()).toBeFalsy();
+    expect(res.status()).toBe(401);
+  });
+
+  test('rejects requests with invalid token', async ({ request }) => {
+    const res = await request.get('/auth/profile', {
+      headers: { Authorization: 'Bearer invalid.jwt.token' },
+    });
+
+    expect(res.ok()).toBeFalsy();
+    expect(res.status()).toBe(401);
+  });
+});