How to check if a category has a parent and child categories?

Question

First of all consider that if the category exists, term_exists function return an array, not a term id. See Codex.

So:

$term_ids = array();

$term = term_exists('categoryone', 'category', 0);

if ( is_array($term) && isset($term['term_id']) ) { // term exists as parent cat
  $term_ids[] = $term['term_id'];
  // get all the children of the categories
  $children = get_categories( array( 'parent'=> $term['term_id'], 'hide_empty'=>false) );
  if ( ! empty($children) ) {
     // category exist as parent and also have children
     // do you want to add also the children? If so:
     $children_ids = wp_list_pluck($children, 'term_id');
     $term_ids = array_merge($term_ids, $children_ids);
  } else {
     // category exist as parent but has no children   
  }
} else { // term does not exists as parent cat
  $insert_term_id = wp_insert_term( 'categoryone', 'category' );
  $term_ids[] = $insert_term_id;
}

And then, if $insert_id is the ID of a post you can use:

wp_set_post_categories( $insert_id, $term_ids );

For more info see codex for