How can I make an order generated through Services invoke hooks or checkout complete?
So, for anyone looking for a solution, here is what I did after asking for help in IRC #drupal-commerce:

Don't use the order status, don't use a checkout-status. Use something (an entity) as a trigger that is necessary for creating the order. I picked the last action provided via .json-file, POSTing a line-item. As soon as this is known by Drupal, all corresponding rules start their work, and some hooks respectively. 

Sidenote: I learned something else also and wished to have been pointed to that earlier: Next time I will not use the above mentioned modules again. They're buggy and obsolete in D8 anyway since the concurring RESTws module became part of core.