Theme/Plugin/Widget Data and Staging/Production Servers

Native WordPress import/export file is focused very tightly on content. It is not meant and cannot (or at least no one bothers to try bend it that way) accurately represent the state of whole site.

The missing storage you are looking for is likely Options API, which is heavily used by core and third party extensions both. It persistently stores arbitrary data into options table in database.

The accurate replication of WordPress site takes complete copy of database. There still isn’t any “default” solution for that in WordPress toolbox and process tends to vary a lot between developers (command line dumps, direct DB to DB copy, deployment scripts automating either, etc).