Is it possible to drag and re-order the folder tabs? If not, that would be a real nifty feature.
Yes, you just drag them. 
Maybe you have turned on Preferences / Folder Tabs / Options / Treat tab label as folder when dragged or right-clicked (which is off by default), in which case you have to drag the icons and not the labels. Otherwise, you can drag any part of the tabs (except the close buttons).
That did the trick! Thank you.