How do I attach documents to a message template? Am I overlooking an "attach" button?
I believe that's not possible yet (it would be a good feature to add to CiviCRM). Adding attachments is only possible sending an email or bulk email, not when you create/edit a message template.
You could upload the file/attachment to your server/site and include a link that points to it in the message template so recipients can download it.
I just stumbled on an extension for this: https://civicrm.org/extensions/e-mail-template-attachments
Although, I'm having a little trouble with it maybe you'll have more luck.
When you do Actions> Phone Call, there is the Attachment(s) section at the bottom of the screen. You can add files here. You can then send an e-mail with the file attachment to the person or mail group you assign to Assigned to. Your Phone Call will go as a message template.