Can I display properties in alphabetical order in refine?

The view uses the default sort_order as specified in your ItemType. 

You can modify ..Innovator\Client\Modules\aras.innovator.ExtendedClassification\scripts\BaseColumnSelectionMediator.js
But I wouldn´t recommend general alphabetical sorting. The current sorting represents the grid order, which is most of the time quite helpful.

Personally I would rather add some additional button to the dialog that offer additional sort options on demand:

1. One button for optional alphabetical sorting
2. Second button to show the original database name of the property (mainly for admins :) ).