Help me in buying best graphics card. Specs: Intel DG31PR, Core-2-Duo 2.8Ghz, 2.

doesnt exist. $50 will get you something that would get about 2 FPS on newer games with max settings and still pretty unplayable with min settings at a 1920x1080,
you can check ebay and see if you can get a 4600, 5600, 6600 series card or better. for that much. i dont know Nvidia that well for midrange.

Have you considered a platform upgrade? What you're using now (775) is old, and with a C2D I would guess even with an OK graphics card in there you probably wouldn't get satisfactory results with newer games. This is all IMO, you can do whatever you want.

I would like to think that a person using a C2D and Intel graphics cant afford a 1920x1080 monitor - so my uneducated guess would be 12x10 or maybe 16x10.

With this system it doesnt really matter. Buy anything that meets your budget, new or used, and it will be better than that Intel graphics. but its not going to be the best. You can game fine but dont expect superb results.

As soon as you plug that new GPU in your need to think about replacing the CPU, mb and the new GPU. The good new for you is so many game developers are targetting the consoles and they play decent enough even on your system.