fix: browser testing issues and missing UI tools

- Fix WebSocket URL to use ws:// for http development environments
- Make createLayer idempotent to prevent race-condition 500 under React StrictMode
- Add URL-based routing for /register so register page actually shows register form
- Add missing tool buttons to LeftSidebar: polygon, chair, extend, fillet
- Add Playwright browser test suite for exhaustive UI testing
- Update .gitignore for Playwright artifacts

Unit tests: backend 254/254, frontend 380/380

frontend/playwright.config.ts

@@ -0,0 +1,35 @@
+import { defineConfig, devices } from '@playwright/test';
+
+export default defineConfig({
+  testDir: './playwright-tests',
+  fullyParallel: false,
+  forbidOnly: !!process.env.CI,
+  retries: 0,
+  workers: 1,
+  reporter: [['list'], ['json', { outputFile: './playwright-tests/playwright-report.json' }]],
+  use: {
+    baseURL: 'http://localhost:5173',
+    trace: 'on-first-retry',
+    screenshot: 'only-on-failure',
+  },
+  webServer: [
+    {
+      command: 'cd /tmp/web-cad-clone/backend && npm run dev',
+      url: 'http://localhost:3001/api/health',
+      reuseExistingServer: true,
+      timeout: 120000,
+    },
+    {
+      command: 'cd /tmp/web-cad-clone/frontend && npm run dev -- --port 5173',
+      url: 'http://localhost:5173',
+      reuseExistingServer: true,
+      timeout: 120000,
+    },
+  ],
+  projects: [
+    {
+      name: 'chromium',
+      use: { ...devices['Desktop Chrome'], viewport: { width: 1600, height: 1000 } },
+    },
+  ],
+});