Ordering documents instead of nodes

I don't think so since NodeOrder is stored in CMS_Tree and not CMS_document.

Its not out of the box feature. As idea, you need custom pagetype property where you define order for items and save settings to custom table

The custom page type property will work for the parent page in which all the children live under but will not work for the children.

Another option is use a localization string. In your Order By box, add the macro to get the key {$custom.OrderBy.Articles$} for your string and manage it in the Localization app. In there you can add your different order by fields based on the culture.