Use the installed headers

This fixes synqt warnings about incorrect inclusions in public headers.
Also deals with extra- and duplicate includes, forward declaration,
namespacing issues, and circular dependencies...
...and Qt-ifies the headers after all.

Change-Id: Ia373d6fb93e0fecedd2a2334fbeb4c2005a67a4c
Reviewed-by: Michael Krasnyk <michael.krasnyk@gmail.com>
Reviewed-by: Alex Blasche <alexander.blasche@digia.com>