When you press Printscreen, it copies the entire screen. Load up a paint program, and paste as new image.
The screenshot key for JJ2 is F12, and it saves it in the JJ2 directory, as a .tga file for 16 bit, .pcx for 8 bit. Jazz***.pcx/tga, *** being numbers.
I don't know about animated Jazz gifs, but you can make them yourself. Holding down F12 long enough will get you an animation, frame by frame.