I had this exact problem...if only I can remember exactly what i did.
and yes all those to all those questions, this is the probably reason. Your computer willl turn itself off if it gets dangerously hot/not enough power.
Check your power connectors, swap some around so your connecting your card with a dufferent connector.
In the meantime refrain from using high profile programs (i.e games, digital video editors)as your graphics card could fry at any moment. I will try and get you a better answer when my Dad gets home as he might be able to remember how we fixed this (and it wasn't ordering a new card)
I have a GeForce Ti4200, one day I spotted the card was hot and the GPU fan wasn't spinning...so I wanted a new card, a GeForce FX5600, installed it and turns out the fan on this one doesn't work either...I can tell the fan worked on the Ti4200 because there is dust circulating around the heatsink and fan...but the new FX5600 isn't spinning at all, it is also new...
Is it the AGP port?
Something I have to set in BIOS?
Probable reason why my counter-strike keeps shutting down?
Probably why I sometimes receive the blue screen of death?
Probably why my computer does random reboots?
I have a 550W power supply, enough juice for everything...anybody know the problem? Please help!