fastest between hook_js vs hook_menu to get JSON via REST
Actually, as the project documentation says, you should use http://example.com/js.php?q=js/mymodule/callback instead of http://example.com/index.php?q=js/mymodule/callback, and basing on the test reported from the maintainer, the previous is faster than the latter (59.371 ms per request versus 446.846 ms per request). Since you are waiting a reply from another server, probably you would not notice that difference too.

For you second question, if you are going to use the High-performance JavaScript callback handler module in the way shown in the documentation, js.php bootstraps Drupal at DRUPAL_BOOTSTRAP_DATABASE level. In any case, if you are not using any Drupal function nor the session, and you are using just plain PHP functions or classes, then you don't need to bootstrap Drupal at all. Keep in mind that this means you cannot use any function implemented by Drupal modules too.