integration: return HTTP 404 for unmatched URLs and close remaining browser-gate gaps

Co-authored-by: Cursor <cursoragent@cursor.com>

tsconfig.json

@@ -28,6 +28,9 @@
     },
     {
       "path": "./tsconfig.spec.json"
+    },
+    {
+      "path": "./tsconfig.e2e.json"
     }
   ]
 }