In the Ektron API, if I have a MenuTreeNode item, how do I determine if the page added to the Menu list is published or unpublished.
When you say Ektron API, I assume you are referring to the MenuManager?
The status of a content item is held in the ContentData object which is retrieved from the ContentManager API.
var contentManager = new ContentManager();
var contentData = contentManager.GetItem(contentId);
var contentStatus = contentData.Status;
In order to get the content ID you can look in the MenuItemData object which can be had from the MenuManager API.
var menuManager = new MenuManager();
var menuItemData = menuManager.GetMenuItem(menuItemId);
var contentId = menuItemData.Itemid;
Finally content which is "unpublished", i.e. does not have a status of "A", will not be retrieved by the APIs shown above. In order to get content that is not published, the APIs need to be told to be in Site Preview mode like so:
contentManager.InPreviewMode = true;