Hi,
Joomla's library does not use Symfony. The main libraries it uses are located in:
libraries/cms
libraries/joomla
libraries/legacy
libraries/vendor/joomla
as of Joomla 3.4 alpha. The symfony library is only there to provide the YAML package, which although isn't used by the CMS itself is an option for importing YAML data into our Registry package (which you can find out more about here:
https://github.com/joomla-framework/registry)
As to the structure of the CMS, the files between the 4 folders mentioned above go as follows. The stuff in libraries/joomla contains largely the defunct Joomla Platform project (
https://github.com/joomla/joomla-platform). Whilst the platform project is deprecated the code within this folder is not
The code in libraries/vendor/joomla is part of the Platform replacement Joomla Framework (more info here:
http://framework.joomla.org/). Many of the platform classes which share code with the framework class are slowly being extended from the framework class (since the coming release of Joomla! 3.4).
The code in libraries/legacy contains code that used to be in the platform but is now considered deprecated. Note there are a limited number of classes in there that were deprecated by the platform project but are NOT deprecated (for example JCategories and some of the JTable instances) but as a good rule of thumb it's deprecated code.
Finally the code in libraries/cms contains code specific to the CMS that is not intended to be used outside of the CMS environment (things like our own pagination class, the class for rendering components etc.).
FOF is a library to rapidly produce 3rd party extensions - and has some classes (like the things for 2 factor authentication) that are in the CMS. However we are moving most of the more useful FOF classes into core and by Joomla 4 this will likely no longer be included with Joomla. FOF is produced and maintained by Akeeba and there is a full set of developer documentation for it here:
https://www.akeebabackup.com/documentation/fof.html.
Hope this helps!
Kind Regards,
George