Outlook 2013’s IMAP implementation has changed quite dramatically when compared to previous versions of Outlook and has remained mostly the same in Outlook 2016.
While it does do a better job at getting the special folders correct more often, there sadly is no more option to change it manually in case Outlook gets it wrong.
As you’ve noticed, folders marked with “This computer only” are not being synched back to the mail server.
When you’ve set your IMAP mailbox as the default data file, then it will also contain Calendar, Contacts, Tasks and Notes folders which only exist on your local computer.
With some IMAP servers, Outlook 2013 and Outlook 2016 will still have issues determining the correct folders.
In the dialog that pops up, press the Query button.
If all folders start with a common name followed by a dot, then that first part is your root name.
Sadly, there is nothing you can do in this case other than reporting it as a bug via Microsoft Product Support.
The Outlook Team is aware of the issue and is also looking into ways to bring back the manual folder mapping dialog or to make Outlook get it right the first time more often.