The brightness is controlled by the hotkey feature of your monitor and it shouldnt' be affected by how the computer is controlled. Only exception is if it was a laptop then they have a special hotkey set that is required to run to control the keyboard shortcuts.
As far as your video card fan it will indeed run all the time and will increase fan speed which can be heard when fairly clear when it's starting to run hot after playing games for an extended period.
Speedfan is a good control to regulate fan speeds of your devices *if supported* some nvidia and ati video cards have their own fan controllers in this case. http://www.almico.com/speedfan443.exe
As far as saying it shuts itself down, I'm not exactly sure of what you ment. Are you saying the video completely blacks out or the whole computer or are you refering to the fan literally stopping rotating which is actually fairly dangerous since a video card can get very hot if it's uncooled properly. *exceeds 60-100c if fan completely dies*
A good way to check this is open up the side of your case and put your hand near it but don't touch it directly and you should be able to get a good idea of if it's running hot or not.
Let me know how this turns out and I'll try to help out some more.