Well, you need a higher resolution monitor if you want to push the GTX 670 to its limits. 1366x768 is a very low resolution, frankly, the GTX 670 is overkill for such a low resolution. Odds are your CPU will determine how games perform more than your graphics card with your current display.
Get at least a 1080p monitor, so the GTX 670's power doesn't go to waste. Even then, the 670 will run those games fine on ultra at 1080p. If you are looking to make your 670 struggle, you'll either have to play Metro 2033 on ultra, or get into ultra high resolution displays (eg. 2560x1600), or multi monitor gaming.
You can plug in both monitors if you want to, though I would recommend only using the 1080p display for games. You can extend games across multiple monitors, but it is recommended you only do that on displays of the same size and resolution, and typically one does this with three displays, not just two.
DVI and HDMI transmit the same video signal, you'll see no difference in picture quality. The only reason to get an HDMI cable would be if your new monitor had speakers, and you wanted to run audio through the monitor as well as video.