To add fields to private messages, navigate to Admin -> Structure -> Private Messages (/admin/structure/private_message/private_message), and add fields as with any other entity.

The order of items in a private message thread can be re-ordered at Admin -> Structure -> Private Messages -> Manage Display [tab] (/admin/structure/private_message/private_message/display). On this page, the message author, creation date, and message can be reordered.