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>

playwright.config.ts

@@ -0,0 +1,62 @@
+import { defineConfig, devices } from '@playwright/test';
+
+/**
+ * Playwright E2E configuration for Goodgo Platform.
+ *
+ * Projects:
+ *   - "api" — tests against the NestJS API (port 3001)
+ *   - "web" — tests against the Next.js frontend (port 3000)
+ */
+export default defineConfig({
+  testDir: './e2e',
+  fullyParallel: true,
+  forbidOnly: !!process.env.CI,
+  retries: process.env.CI ? 2 : 0,
+  workers: process.env.CI ? 1 : undefined,
+  reporter: process.env.CI
+    ? [['html', { open: 'never' }], ['github']]
+    : [['html', { open: 'on-failure' }]],
+
+  use: {
+    trace: 'on-first-retry',
+    screenshot: 'only-on-failure',
+  },
+
+  projects: [
+    // API E2E tests — no browser needed, uses APIRequestContext
+    {
+      name: 'api',
+      testDir: './e2e/api',
+      use: {
+        baseURL: process.env.API_BASE_URL ?? 'http://localhost:3001',
+      },
+    },
+    // Web E2E tests — Chromium browser
+    {
+      name: 'web',
+      testDir: './e2e/web',
+      use: {
+        ...devices['Desktop Chrome'],
+        baseURL: process.env.WEB_BASE_URL ?? 'http://localhost:3000',
+      },
+    },
+  ],
+
+  webServer: [
+    {
+      command: 'pnpm --filter @goodgo/api run dev',
+      url: 'http://localhost:3001',
+      reuseExistingServer: !process.env.CI,
+      timeout: 30_000,
+      env: {
+        NODE_ENV: 'test',
+      },
+    },
+    {
+      command: 'pnpm --filter @goodgo/web run dev',
+      url: 'http://localhost:3000',
+      reuseExistingServer: !process.env.CI,
+      timeout: 30_000,
+    },
+  ],
+});