Read the product detail that explain the sharing users and the single sign-in.
www.jms2win.com/download
In summary, you can create several website that share the joomla users or share some extension like community builder or jomsocial.
There is a limitation concerning your environment. You have to check that you are using MySQL 5.0 and that you have all the MySQL privileges to share the MySQL tables.
See tutorial video 16 that show how to share users.
www.jms2win.com/tutorial#jms12x-16
See also the tutorial video CP-02 that show how to create a DB on cPanel and limitation concerning the "MySQL" privileges.
www.jms2win.com/tutorial#jms12x-CP-02
Concerning the single sign-in it only works on the same domain.
This mean that your websites can be subdomain or subdirectories.
Concerning ELGG and Moodle we don't know.
Can you give send us these joomla extension to allow us review them.