The free core template do show the categories within the details but none of the premium templates do for two reasons. The first most users seem to prefer not to show that and second that many of the premium templates have a method to filter by category. If the user is opting to filter by a category, they will now the category of the entries being shown making it redundant info.
So all three membership levels are WordPress user levels, yes … nothing really to do with Connections other than they should be able to view the directory or not right? If that is the case, you can set the role capabilities on the Connections : Roles admin page.
re: I see the private and public but not sure where the private is referring from….a WordPress user level of permissions?
Public/private can mean two things depending if you have the directory set if the user must be logged in to view it or not.
If the user is not required to login to view the directory, default, then private means the user must be logged in AND have the role capability to view private.
If the user is required to login to the view the directory, nothing will be shown until the user is logged in. After the user is logged in the user will only be able to view entries if they have the role capability to view public and/or private.
I hope that is clear enough and makes sense!
Connections does not integrate with the core WP search for two reasons. Commingling the search results with blog post/page results can be very problematic in displaying the results in a way that is coherent and would make sense to the user. There is also a technical difficulty because Connections does not use a CPT as do many/most/all the other directory plugins. The reason for this is that Connections is older than the CPT feature in WP. I did consider switching to CPT but decided not to as the db schema for post/pages and directory isn’t ideal. It can be made to work like other directory plugins but I would have to drop many of the unique features Connections has vs the others.
Hope this helps!