After reading the comments, I decided that I would try updating the .mflil file.
I went to http://patchup.info/downloads/ to download the file.
Then I added the file to my wow directory and followed the instructions on this page: http://patchup.info/updating/
Once I did what it said and ran the Blizzard Updater. Once updated, I fired up the WoW application itself and logged in. It told me to download and apply a patch. Once it updated the patch, I could log on and play just as normal.
However, the Launcher STILL does not work correctly. Not sure if I missed a post on that bit, but the game is working correctly for me.