Completing the checkout process event isn't firing
Debugging is the art of eliminating possible causes for an issue. In this case, your issue could be because ... your rule is disabled.

As per the 'I've disabled all of the rules bound to the event "Completing the checkout process" ...', double check that the rule you want to fire is (still) enabled. If it isn't, then make sure to enable it (again).

Also, your question states "the e-Mail is never send": it could well be because something (different from your rule not firing) is wrong. So therefor, add an extra Rules Action to the rule you're struggling with, as the very first Rules Action, and which is only to "show a message on the site"", and with a message like "Bingo" (or whatever). Then rerun your testcase, and verify if you do get such "Bingo":


If you do, your rule has fired ... and your real issue is related to sending the eMail.
If you don't, (temporary) remove ALL Rules Conditions in your rule and retry. If in that case you do get a "Bingo", then your issue is related to one or more of your Rules Conditions (though the rule did fire).


The above is how I typically do most of my Rules debugging, in cases where some rule seems to not fire (stupid, but pretty effective).

If none of the above debugging tips helps ... then edit your question and include an export of the rule you are debugging (in that case I might further improve my anwer).