Does WordPress have something like Drupal’s DB API?

Yes, there’s the wpdb class. Some examples can be found for e.g. here.

Then there’s the global $wpdb object that contains results and other parts, set based on the current routing.

You can also utilize the WP_Query class which is more common. There’s a default global $wp_query object and the $wp_the_query(?) intermediary object, that holds the default routing based result. You can use WP_Query basically like any old school query builder and call it a poor mans ORM.