What do you mean by "define custom proporties as the default choice for all documents"? There is no setting in yEd that prevents you from creating custom properties for your diagrams.
Moreover, the only "Apply" related to custom properties I can think of is the properties mapper's "Apply" button. However, this button does not define anything but executes the chosen transformations for your diagram. And no, there is no way to automatically run these transformations.
