As far as I can google, WordPress offers no convenient way for accessing some things I'd like to use.

wordpress_category_functions.php

<?php

// Utilize wordpress' get_queried_object to get the details necessary.
function current_category_ID()
{
  $category = get_queried_object();
  return $category->term_id;
}

// Boolean: Does this category have child categories.
function category_has_children($category_ID)
{
  $test_word = 'THIS_IS_MY_TEST_FOR_NO_CHILDREN_LA_LA_LA_LA_LA';
  $children = wp_list_categories(
                array(
                  'child_of' => $category_ID
                , 'echo' => false
                , 'hide_empty' => false
                , 'hierarchical' => true
                , 'show_option_none' => $test_word
                , 'style' => 'none'
                , 'title_li' => ''
                ));
  return !($children == $test_word);
}