Most recent post from every category/taxonomy but sorted by date

In your code – Find

$cat_args = array(
    'type' => 'post',
    'orderby' => 'name',
    'order' => 'ASC',
    'taxonomy' => 'tax_name');
$categories = get_categories( $cat_args );

Repalce

$cat_args = array(
    'type' => 'post',
    'orderby' => 'date',
    'order' => 'DESC',
    'taxonomy' => 'tax_name');
$categories = get_categories( $cat_args );