Is there a way to automatically do this?
 

I have a green screen, and it's all working well now, but I still want one more thing. How can I make it so that when I zoom or move, the background picture also moves?

You could try a software solution such as Boris Red or After Effects, and use corner pin tracking of the footage to create a track you apply to the background track.You might need to set something up in the scene that you shoot the foreground footage in to track to.
You could also manually track the zoom in the foreground to the background if you don't do a lot of zooms.(using your NLE)

I know you can do pans with AE using the motion tracker as long as you have some markers on your green screen you can track. Zooms can be a little trickier I imagine, but they can be done.