I received your sample PDF, and the diagnostics tool output, and the most evident error is that you have the custom fields differently named in the PDF and in the PDF-ShellTools configuration. I suppose you used PDF Explorer to create the fields, and in PDF-ShellTools you also created the fields manually, instead of importing them.
What happens is that PDFE uses PDFECustomX as default name for the fields, and PDF-ShellTools uses PDFCustomX, so PDF-ShellTools is trying to read different fields from your PDFs.
To fix it, open the PDF-ShellTools manager and change the names to PDFECustomX. See first attached image.
This change field name operation, if done from the basic custom fields settings editor (don't happen if done from the advanced settings editor), will make the fields mapping to still refer the old name (due to a bug that will be fixed in the next release), so you also need to change the field label or use the mapping tab to map the field to another system defined property. In your case it's even what you should do, because one of your custom fields is named Kommentarer and the Windows property system already has a "comments" property defined, used by files that natively have this metadata field, such as MS Office documents.
So, let's map your Kommentarer field to the system.comments property. This way you can have the column comments showing metadata from PDFs, along with other file types that also populate this field.
Just use the property handler mapping tab to map your field to the system.comment one (check second attached screenshot for better reference).
The label of the system field may be different (than "Comments" as in the screenshot) if your Windows UI uses a different language The important is to find the property with System.Comment as the [canonical name].
Now check under the details pane items tab if the fields are still checked, and apply the changes.
When setting the Kommentarer field as visible column in Windows Explorer, and if your Windows UI language is also Danish, you may have two fields named kommentarer (I have no idea if the Danish Windows uses "kommentarer" for the system.comment property). If that's the case, the only way to know the correct one is to make visible the two and check the one that shows also the comments for MS office files too. In this situations it is better to label the custom fields with a different name, such as "PDF kommentarer", so you can better distinguish your fields from the system ones.