Apologies for the delay in my reply.
Since you have three categories set up and you want three menu items the best option is to use the category shortcode option. Create three new pages, put the Connections shortcode on each using the category shortcode option set with the appropriate category ID number.
To setup the viewing as you described … go to the Connections : Settings admin page and under the Display tab check the Single Entry option. Now go to the Connections : Templates admin page and activate the Names template under the All template type and under the Individual and Organization template type, activate the cMap template. That should do it.
Hope that helps.