You can run Doom 3 with virtually any "newer" gpu. By newer, I don't just me a GeForce 6 or the ATI X8*0s. Most people consider Doom 3 to be the most beast graphics demanding game out there, and it is.
However, you can easily play Doom 3 on low quality with almost any GPU. I was able to push 1024x768 High Quality on my 5700 LE very easily, running an acceptable average framerate of 40.
You should definitely be able to play Doom 3 with a 6200, but don't expect some fantastic quality settings. BF2, I think that may be a bit far out of reach.
Keep in mind that many modern games are processor bound. You can get a ridiculously beast GPU, but if you have a lousy processor, then it's all gone to waste.