Is there any way to move and copy nodes in one action?

In Photoshow, when I select an object and move it while holding alt, it automatically creates a duplicate of the object and moves the object beneath my cursor. So it's basically two actions in one:
Move + copy

Does something similar exist in Fusion for nodes?

No. But you can copy (Control/Cmd-C), then click where you want the duplicate node to go, then paste (Control/Cmd-V).



(If you wanted to copy and move the position of something in the viewer, ie change where a node is in the actual frame, then no method exists for that that I know of.)