Hi Robyn,
First off, what version of the Laserfiche Client are you using? There's no incompatibilities with Office 2016 in current versions of the Client so lets confirm that first.
It sounds like you may have identified the export issue, but just in case - my guess is that you're dragging the file out with one or more of ctrl or shift selected, which drags them out as a shortcut. Normal drag and drop shouldn't do that without modifiers, so that'd be what you'd want to check. That may also happen if you don't have the export feature right anymore for some reason so that'd be another thing to look at.
For import, we'd really need a bit more details as to what exactly type of failure you're getting. Does it work on any of the ways to import emails, or does it fail on all of them: File\Import from the desktop, drag/drop from the desktop, drag/drop directly from outlook, send to laserfiche from the outlook ribbon? Is there a specific error message? Do the documents get created without content, or does it just fail outright? Is this specific to just emails - can you import other office documents without error? (I know it's a lot of questions, but it's hard to tell what's going on without this sort of information).