Menu Closing alone

Well, I think that's it, your script starts by checking the " .nav-toggler ", so if it was clicked, it adds the class " .open " to the element with class .menu . script, understand this as a computer processing the script. the next block detects again " .menu ", then detects a click on the element " body " which is the body of the page, if it was clicked it removes the class " .open " from " .menu " , and I think that's your problem. because " .nav-toggler " is inside the body tag, hence the script while adding the class it removes next. is resolved only by removing the block:

jQuery('body').on('click', function (e) {
  jQuery('.menu').removeClass('open');
});

Instead, use another technique to detect the click off the menu: link

You do not need to remove anything from the code and keep the click functionality on body to close the menu. Leave your code as is and add% as_with% as below.

Your problem is in if in this part of the code that simulates a click on the page, closing the menu:

// Carousel on Tabs of 'Unidades de Lazer'
tabCarousel = setInterval(function(){
   var active = tabNavItem.filter('.active'),
   next = active.next('li'),
   toClick = next.length ? next.find('a') : tabNavItem.eq(0).find('a');
   toClick.trigger('click');
}, 3500);

Add a trigger to check that the menu is open by preventing it from being closed by itself:

// Carousel on Tabs of 'Unidades de Lazer'
tabCarousel = setInterval(function(){
   var active = tabNavItem.filter('.active'),
   next = active.next('li'),
   toClick = next.length ? next.find('a') : tabNavItem.eq(0).find('a');
   if($('.nav-toggler').hasClass('open')){ // este if
      toClick.trigger('click');
   }
}, 3500);