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-register.spec.ts

@@ -0,0 +1,52 @@
+import { test, expect } from '@playwright/test';
+import { createTestUser } from '../fixtures';
+
+test.describe('POST /auth/register', () => {
+  test('registers a new user and returns token pair', async ({ request }) => {
+    const user = createTestUser();
+
+    const res = await request.post('/auth/register', { data: user });
+
+    expect(res.status()).toBe(201);
+    const body = await res.json();
+    expect(body).toHaveProperty('accessToken');
+    expect(body).toHaveProperty('refreshToken');
+    expect(typeof body.accessToken).toBe('string');
+    expect(typeof body.refreshToken).toBe('string');
+  });
+
+  test('rejects duplicate phone registration', async ({ request }) => {
+    const user = createTestUser();
+
+    // First registration should succeed
+    const first = await request.post('/auth/register', { data: user });
+    expect(first.ok()).toBeTruthy();
+
+    // Second registration with same phone should fail
+    const second = await request.post('/auth/register', { data: user });
+    expect(second.ok()).toBeFalsy();
+    expect(second.status()).toBeGreaterThanOrEqual(400);
+  });
+
+  test('rejects registration with missing required fields', async ({ request }) => {
+    const res = await request.post('/auth/register', {
+      data: { phone: '0912345678' },
+    });
+
+    expect(res.ok()).toBeFalsy();
+    expect(res.status()).toBe(400);
+  });
+
+  test('rejects registration with short password', async ({ request }) => {
+    const res = await request.post('/auth/register', {
+      data: {
+        phone: '0912345678',
+        password: 'short',
+        fullName: 'Test',
+      },
+    });
+
+    expect(res.ok()).toBeFalsy();
+    expect(res.status()).toBe(400);
+  });
+});