xircon, as I suspected, the issue isn't at all related to that package, it's related to installing any package, in this case, the command is reinstall, not install, so it's installed, it gets reinstalled.
The install is what is failing, same happened above in the first session in your log file.
I assume that you are using wifi, this is my guess, and that you are using the mint wifi connection manager, which may not properly enable wifi connection out of X, though why the ftp downloads and sgfxi update work I can't say.
This is a problem in Mint almost certainly, not in sgfxi, sgfxi is simply stating a fact, the apt-get install command exited with error.
I should add some more logging in there I think to show me what runlevel the user is in, if X is actually running at the time, and a few other things that might help me pin down issues like this.
However, there's no problem with sgfxi, it's all working correctly from what I can see, the error message is exactly right, there is a problem with apt.
You might have a borked apt-get that shows error every time it's used, I can't really tell you.
You can test yourself: apt-get install --reinstall any-package
glx may not be reinstallable if X is running, I can't really say.
I do see space for more logging of system data here though.