My problem in Outlook 2013 is that when I set up my Folders in Favourites and put them in an Order which suits me I find that this Order is lost after closing down the Application.

In other words how do I prevent my Folders in Favourites from moving from the Order I have placed them?

These folders probably apply either to IMAP folders or online (non-cached) Exchange mailboxes then?

This is indeed a bug in Outlook related to timing. There is nothing that you can do about this aside from contacting Microsoft Support and opening a bug support case.