Is it possible to “download” a PDF from SharePoint using Filemente, hand over the document to the PDF editor, and then put it back into SharePoint?

You can use “Open In…” menu in Filamente to send a PDF file to other applications such as iAnnotate. Once you finished modifying the PDF file from iAnnotate, you can send the file back to Filamente using “Open In…” menu in iAnnotate. The modified file will be stored in “Local Documents” section in Filamente, and you can upload the modified PDF file to SharePoint via “Upload from Local Docs” menu.