Is my Graphics Card Dead?

I ordered a new PC recently, 500GB Memory, 2GB RAM, 650W PCU, and a GeForce 8800 Graphics Card, all on Vista 64-bit.

However, I have been having constant crashes in any games I try to play. M2TW has random crashes which sometimes give an error message saying the game has stopped running properly. RTW won't even start, although I think that is just an installation problem. In Battle for Middle-Earth, the game crashes fairly regularly and sometimes gives an error message, and sometimes freezes completely. The most recent problem is that upon loading a save game, the screen gives a bright multicoloured dislplay of pixels and big triangles and squares all over the screen.

I have no idea whats causing these problems. Could it be that the graphics card is faulty?

Well, when that happens for me there might have been a corrupted file somewhere, but given its consistency, I suspect that maybe your videocard is overheating??. I dont know too much about power sources, but you should just post it for other people

Is your graphics card overclocked? It sounds like you maybe getting artifacts which happen when the graphics card card is overheating.
Daft as it sounds, I'd open the case and sit a fan blowing onto the card, see if that helps. If it does, get a good case fan.
Or else try backing off the core and memory speed.

I'm not overclocking it at all. Also, some of these graphics problems occur as soon as I start a game up, so it shouldn't be overheating by then. The card already has its own little fan, as-well as four other main fans.

Have you checked your drivers?

All cards have their own "little fan", doesn't mean it's not overheating. And just because you haven't OC'd it doesn't mean it's not OC'd as stock.
Without adequate ventilation and in the wrong environment, it's entirely possible that a standard card could be running too hot.
But as has said drivers may help too.
Just sounds to me like overheating, do you realise how fast a card or processor heats up? If your CPU isn't cooled right, it has a lifespan of about 10 milliseconds. Your GPU wont be too far behind.

Sounds more like some kind of bug. Have you had your card checked. maybe its defective. What model 8800 do you have. I have one and it wont play VI for instance.

Gawain makes a fair point. I don't know very much about anything specific (manufacturer/card specific). I wouldn't know the difference between a 7900 and an 8600 pertformance wise.
I only know a little about general issues that are relevant to any setup, and I would first update the drivers and then cool it as best you can. See if it helps.
Something else, has it been working fine for some time or is this something that has happened ever since you had this system.