No. Resolution DOES affect performance. The higher the resolution, the more data the video card has to process. Color depth matters too.
The number of physical pixels on any given monitor (LCD anyway) is constant, but the number of pixels that are used to represent the computer interface is very much variable.
This is really pretty easy to see. Take any random game that doesn't use a fixed resolution. Run it at say 640x480 and measure FPS, then run it at whatever the highest resolution is your monitor supports and measure FPS again. You should see a noticeable dropoff.
More then likely not
Resolution is basically a format to fit the screen this does change picture quality determinate upon the size of the screen but does not change the pixel density (number) of the image, this remains fixed. Since graphics card performance is determinate on the creation of pixels on the screen and since the number stays the same performance should remain relatively constant.
Is this correct? If not what is the correct answer and why?