I have a gateway FX laptop with 1400 x 1200 screen resolution. Can I swap this screen to another FX laptop with a current screen resolution of 1400 x 900 if they have the same video card or will there be other consideration to deal with?

I recall doing this with Dell models. To see if it works on the Gateway I think you'll just have to try it. Dell's designs were such that you changed the entire top lid (that part with the LCD) and you could get a higher or lower resolution. Just the lid component, no CPU change.