Hi Evan,
Yes, as per the instructions for upgrading to Office 365 2013, we installed Lync 2013 Basic and tried to run the Desktop Setup.
I downloaded and installed IE10 on my PC yesterday afternoon. No change.
However, some additional information that I discovered, this problem occurs in our "Public" site
It does not happen in our "Engineering" site.
In the "Engineering" site, if I click on a Word or Excel document it opens in the desktop client (Word 2010 or Excel 2010). It doesn't check the file out (unless we do that as a separate initial step), but it is opening the correct way.
In the Public site we get the behaviour I described in my first post of this thread. Playing around with it I have found that in this site I can check the file out (on the server) and edit, or view, in the web app, which I didn't think we had with an E1 setup.