How to make the "Height" of Views Slideshow Fully responsive?
I normally take care of the height with javascript.  First I make sure that I am handling the width properly with CSS:

.views-slideshow-cycle-main-frame,
.views-slideshow-cycle-main-frame-row,
.views-slideshow-cycle-main-frame-row img {
  width: 100% !important;
}


Then I create a function that sets the height.  This will take whatever container your slideshow is in as a parameter and find the slideshow within it.

/* Slideshow resize function */
Drupal.theme.prototype.slideshowResizer = function (target){
  var slideheight = 0;
  $(target + ' .views-slideshow-cycle-main-frame-row').each(function(){
    slideheight = $(this).find('.views-slideshow-cycle-main-frame-row-item').innerHeight();
    if(slideheight != 0){
      $(target + ' .views-slideshow-cycle-main-frame').css('height',slideheight+'px');
      $(target + ' .views-slideshow-cycle-main-frame-row').css('height',slideheight+'px');
      return false;
    }
  });
};


And finally I call the function on the window resize event:

Drupal.behaviors.slideshowResize = {
  attach: function (context, settings) {            
    $('.your-slideshow-container', context).once('processed', function () {
      Drupal.theme('slideshowResizer', '.your-slideshow-container');
    });

    $(window).resize( function() {
      Drupal.theme('slideshowResizer', '.your-slideshow-container');
    });         
  }
}; 


Also, make sure your base img tag has height: auto;