The Intel driver manager will not work on Mint, even though Mint is Ubuntu based.
However, you still can install the drivers without too much manual work, see:http://www.webupd8.org/2013/04/how-to-use-intel-linux-graphics-drivers.html
But take care!
There's no guarantee that it will work flawlessly, apparently some users had issues, and removing it again is a bit difficult.
I used it and for me it mostly works, although I suspect that one bug I get is caused by them (I don't see the wallpaper when switching desktops/using expo mode, only a black desktop).
Also, your problems might be caused by driver issues, but they might be independent. You can find a lot of topics about brightness keys not working, and solutions that might fix it without installing a different driver.