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/fixtures/auth.fixture.ts

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+import { test as base, type APIRequestContext } from '@playwright/test';
+
+/** Shape returned by POST /auth/register and POST /auth/login */
+export interface TokenPair {
+  accessToken: string;
+  refreshToken: string;
+}
+
+/** Generates a unique test user payload for each test run. */
+export function createTestUser(suffix = Date.now()) {
+  return {
+    phone: `09${String(suffix).slice(-8).padStart(8, '0')}`,
+    password: 'Test@1234!',
+    fullName: `Test User ${suffix}`,
+    email: `testuser${suffix}@goodgo.test`,
+  };
+}
+
+/** Registers a new user via the API and returns the token pair. */
+export async function registerUser(
+  request: APIRequestContext,
+  user = createTestUser(),
+): Promise<TokenPair & { user: ReturnType<typeof createTestUser> }> {
+  const res = await request.post('/auth/register', { data: user });
+  if (!res.ok()) {
+    const body = await res.text();
+    throw new Error(`Register failed (${res.status()}): ${body}`);
+  }
+  const tokens: TokenPair = await res.json();
+  return { ...tokens, user };
+}
+
+/** Logs in an existing user and returns the token pair. */
+export async function loginUser(
+  request: APIRequestContext,
+  phone: string,
+  password: string,
+): Promise<TokenPair> {
+  const res = await request.post('/auth/login', {
+    data: { phone, password },
+  });
+  if (!res.ok()) {
+    const body = await res.text();
+    throw new Error(`Login failed (${res.status()}): ${body}`);
+  }
+  return res.json();
+}
+
+/**
+ * Extended test fixture that provides a pre-authenticated API context.
+ *
+ * Usage:
+ *   import { test } from '../fixtures/auth.fixture';
+ *   test('my test', async ({ authedRequest, testTokens }) => { ... });
+ */
+export const test = base.extend<{
+  testUser: ReturnType<typeof createTestUser>;
+  testTokens: TokenPair;
+  authedRequest: APIRequestContext;
+}>({
+  testUser: async ({}, use) => {
+    await use(createTestUser());
+  },
+
+  testTokens: async ({ request, testUser }, use) => {
+    const { accessToken, refreshToken } = await registerUser(request, testUser);
+    await use({ accessToken, refreshToken });
+  },
+
+  authedRequest: async ({ playwright, testTokens, baseURL }, use) => {
+    const ctx = await playwright.request.newContext({
+      baseURL,
+      extraHTTPHeaders: {
+        Authorization: `Bearer ${testTokens.accessToken}`,
+      },
+    });
+    await use(ctx);
+    await ctx.dispose();
+  },
+});
+
+export { expect } from '@playwright/test';

e2e/fixtures/index.ts

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+export { test, expect } from './auth.fixture';
+export { createTestUser, registerUser, loginUser } from './auth.fixture';
+export type { TokenPair } from './auth.fixture';