I don't know if this is the actual app that you want to create or a part of your game but.
You want to create a scene, add a popup(this is where you will add your menu) and script it like this -- 'if you click the right mouse button, you want to set the popup position to your global mouse position and then do a popup(). This will manage the mechanics of right click.
for the object etc., Create different buttons that represent the objects you want to add. and add it to the popup. After that, each button has an export (PackedScene) of your object and then if you click that button you will just instance that object into your scene. and add it as a child. done