WordPress as CMS: How to manage/handle images assigned to a post as full-sized background images in a slideshow?

You can do that within the loop using wp_get_attachment_image.

Something like (not tested):

//loop starts

//values for get_posts ( or wp query if you prefer)
$args = array(
   'post_type' => 'attachment',  //gets the attachments aka your images
   'numberposts' => -1,
   'post_parent' => $post->ID
  );

// some attributes you want added to each image   
$default_attr = array(
   'class'  => "background-attachment", //for your javascript
   'alt'   =>  'something',
   'title' => ''something
);

//loops through the attachments,
//uses the 'full' parameter assuming you want the default large image
$attachments = get_posts($args);
if ($attachments) {
    foreach ($attachments as $attachment) {
         echo wp_get_attachment_image($attachment->ID, 'full', false, $default_attr);
    }
}

//loop ends
//your javascript for full page background via wp_enqueue_script