They do not inherit the parents template.
WordPress has some neat ways to make special templates for certain categories or custom post types (taxonomies). You can apply this to all categories, certain categories by ID or slug and archive pages. The sad part is it does not work in a hierarchical way, you have to make a page for each category..
You can read all about it and how it works on this page: http://codex.wordpress.org/Category_Templates
If you scroll to the bottom of that page you will also find a “Related” section where you can find out how it works for pages, tags, custom taxonomies, author templates and such.
Note: Also check out http://codex.wordpress.org/Template_Hierarchy
BUT there is a fix for this.
You can use this code in your functions.php (or make a plugin)
add_action('template_redirect', 'inherit_cat_template');
function inherit_cat_template() {
if (is_category()) {
$catid = get_query_var('cat');
if ( file_exists(TEMPLATEPATH . '/category-' . $catid . '.php') ) {
include( TEMPLATEPATH . '/category-' . $catid . '.php');
exit;
}
$cat = &get_category($catid);
$parent = $cat->category_parent;
while ($parent){
$cat = &get_category($parent);
if ( file_exists(TEMPLATEPATH . '/category-' . $cat->cat_ID . '.php') ) {
include (TEMPLATEPATH . '/category-' . $cat->cat_ID . '.php');
exit;
}
$parent = $cat->category_parent;
}
}
}
In category page this look for ‘category-XX.php’ template and if it doesn’t exist it looks for the parent cat’s template and so on. And if no category template to the top level cat, then it goes back to the normal template hierarchy.