How to create a array that contains all of the children slugs of a specific parent category

You can use get_terms() to get an array of child slugs:

/**
 * Get child term slugs from given term parent
 *
 * @param Integer $parent_id ( Parent Term ID )
 * @param String $taxonomy ( Taxonomy Slug )
 *
 * @return Array( $term_id => $child_slug )
 */
function get_child_slugs( $parent_id = 0, $taxonomy = 'category' ) {
    return get_terms( array(
        'taxonomy'  => $taxonomy,
        'parent'    => $parent_id,
        'fields'    => 'id=>slug',
    ) );
}

This should return a simple array of array( $term_id => $term_slug ) assuming a valid parent term ID and valid taxonomy is supplied.