If I create a text edit control with NoesisGUI and run it on iOS or Android can I edit and select the text directly in the text box like it's a native control? Or do I get another textbox to edit in right above the keyboard like happens when you use Unity's GUI text box?

Yes, the behavior is the same. In fact we are using the VirtualKeyboard provided by Unity to handle this.

We didn't have time to investigate if this can be improved but I am afraid it won't be possible. To show the virtual keyboard in mobiles you need a native control, problem is that Unity applications are 100% rendered by the GPU, the OS knows nothing about the content. That is the reason a native textbox is shown above the keyboard.