Failed to start game (app already running)...I looked online and found a number of potential fixes, none of which worked for me, including the ill-advised "run Steam as an administrator." -_-
Thankfully, someone mentioned that they had also been using the graphics enhancement program SweetFX before getting the error. Removing all of the SweetFX files cleared it up for me, as well.
Here is a list of all of the files used by SweetFX (found in the Borderlands 2 Binaries\Win32 directory, in my case):
d3d9.dllMove/delete these files/folders from your executable's folder and you should be set.
and the SweetFX folder