Would a higher shutter speed give better edges to pull a key ?

I you're experiencing motion blur, then sure, a higher shutter speed will help clean it up.

Other things that help clean the edge by reducing green spill on the subject:
More backlight/rim light. Then some more!
Increased distance between the subject and the screen
Magenta (minus green) gels on the back/rim lights.
Reduced lighting level on the screen.

Thanks for the tips Seth.