The item is really only in the View menu so that a shortcut key can be applied to it. There is no way to keep it open - OS X menus don’t work like that, I’m afraid.
Because it is not a window. The Window menu is for viewing windows. The View menu is for viewing other things or setting up the way views look
Ultimately, the combination of the “Show Script Elements Menu” and the numbers in the menu itself are intended to work like this:
You hit cmd-Y to bring up the menu and then hit the number of the element you want to change to. E.g. cmd-Y followed by 1 would change the element to Scene Heading.