I need to create a dropdown menu that shows under a button when the button is pressed, and contains a list of clickable buttons. I am struggling to find a way to make a node that can extend itself downards and cover nodes that are below it in the node tree. How would you do this?

Well, as it would just so happen, Godot has this exact node built into the engine already.

It's called MenuButton and does exactly what you describe. It's a button that shows a dropdown menu with a collection of clickable buttons when pressed.