let me ask you this:
in your email templates here
[your-domain]/wp-admin/edit.php?post_type=wppizza&page=templates
you have not actually enabled the post/zipccode output anyway (in your case named as “Ort/Stadt”)
did you just recently disable that ?
This must be enabled to show up in emails to the shop and in the order history

>and an extra order form field (just a simple select field for testing without the zip code plugin).
sure , you can do that , but this will not make any difference to the post/zipcode plugin field being displayed really – it’s just another formfield like name, email etc etc

thing is, i cannot see any of your postcode plugin settings and/or what exactly is enabled and how (if at all)
maybe you can post a screenshot of the settings