Hi Willem,
When you check out a file for editing, if you do not want to keep the changes that you made and want other people to be able to make changes to the file on the server, you can discard the check out. More detailed about Document Connection, please refer to this:http://social.technet.microsoft.com/wiki/contents/articles/10527.sharepoint-working-with-documents-stored-in-a-sharepoint-site-using-document-connection.aspx
For the problem, I have searched other resources, some mentioned that we can only delete files on the web, not from the Document Connection. But I suggest you post the question in the Office for Mac forum to get more advanced and specific information. Our engineers who are more specialized in Office for Mac client issues will help you solve the problem more efficiently. For your convenience, here is the link of the Office for Mac forum that you can click: Office for Mac
Thanks,
Elaine Hao