Nexus theme slider is not showing up when logged out
Yes, it is because your custom.js is causing conflict with the slider.

You need to use the proper synthax for jQuery. 

As mentioned in Working with JavaScript and jQuery

(function($) {
Drupal.behaviors.myBehavior = {
  attach: function (context, settings) {

    //code starts
    $('#main-menu &gt; nav &gt; ul').mobileMenu({
    prependTo:'.mobilenavi',
    groupPageText: Drupal.t('Main'),
    topOptionText: Drupal.t('Select a page')
    //code ends

  }
};
})(jQuery);