The forum has a closed part only for registered users.
i've not well understand this, so i assume something like: there are forums that need to display only for certain groups, but without linking users, nothing display?
The corresponding setting to show the last posts to only registered users is activated.
you refer to option:
Retrieve posts on Last Topics Widget based on phpBB user's permissions?
I've try out in various ways (even if on test i've not setup any particular hidden forum in phpBB) and i've not been able at moment on fly to reproduce this bug.
About to display the number of new / unread post on the forum in the menu:
Yes! You can display any phpBB info retrieved by the query that get logged user's data. The query into class.wp.w3all-phpbb.php (verify_phpbb_credentials() function) retrieve data for the user by all these phpBB db tables:
users
sessions
groups
profile_fields_data
banlist
and could be added more if never needed. This is via php/sql.
For the iframe, any phpBB outputted var, also about template, can be passed to the WP iframed page via ajax, through the overall_footer.html code that need to be added for iframe, and that with easy can be modified to fire any phpBB event, which list (maybe partial) is here:
https://wiki.phpbb.com/Global_Template_Variables