Vues :

In the Web Client you can change the view for document trays and file cabinets to index cards view.
This view shows the thumbnail of the document next to the first 5 index entries. The shown index entries are not relevant for the user.

Is it possible to modify the order?

Yes. First of all switch to Table view. You can change view over the "gear" button:

Change the column order as desired with the mouse by drag and drop .
Now switch back to the index cards view. The changes are applied directly. A restart of the Web Client is not needed.