Is there a way to use Display Suites to change a Layout for a Node Edit/Add form? i.e. form at: .../node/add/
That is possible yes. Display Suite contains a submodule called 'Display Suite Forms' which allows you then to select a layout and move the fields around in that layout.
Note that it's impossible though to move the individual elements in the vertical tabs. That will always remain the same and will rendered at the bottom of the layout.