What exactly do you mean by "moving background"? Are there elements in the background that are moving around, or are there multiple backgrounds that you're trying to switch between? If you're trying to scroll through the same or different backgrounds, you may want to google how to construct one or more crankies (which are often used in puppetry performances). Here's a simple example and more complex example.
I'm also not sure what you mean by:
Is there a way to track hands moving objects objects and cut them out from a surface to add them to an image? I dont know if this can be done in eyes ++, but if it is possible, then I can spare a videoscreen in the background and a camera pointing on to that - because it seems like a bit of a clumsy setup - but this is so far my imagination.
I understand that you are trying to remove the person's hands from the image, but are you trying to move a puppet around in front of a physical background, or a projected background? One possible solution could be to put the puppet on a stick shadow-puppet style, or attach fishing line in two places to the puppet and then hold both tightly outside the camera's view so that you can move the puppet around that way. If you did either of those you wouldn't have a hand in the shot at all. Another possible solution is to have two cameras, one that is aimed at your scenic background, and another that's aimed at a small, well-lit greenscreen with your performer wearing a glove of the same color (you could also try another color like blue, red, or yellow though since there's green in your character). Then you pull both video feeds together and use two different Projector actors with 'blend' set to 'transparent' and the 'layer' input for the character's Projector actor set to a higher number than the 'layer' input on the background's Projector actor.
Here's an example file for that: >>>Greenscreen Example File<<<
There's also a second Scene in that file that should show you how to do the same two-video feed combination workflow, but with a black background behind the character and your performer wearing a black glove.
Obviously, you'll need to tweak values and things to get either Scene to work with live video feeds but I included some Shapes and Text Draw actors to act as stand-ins that demonstrate how the background and character video feeds are combined in both Scenes.
If I misunderstood what you're trying to do, feel free to explain in greater detail.