Really weird as I've never heard of or experienced such a thing. Try creating a new usergroup and assigning that to them and see if it has the same affect.

The parent should be Registered. Creating a new parent of Public has no relation with anything so CB does not know how to use it.

Please PM backend super administrator login credentials and will take a look. Please include exact details to duplicate the issue. Note users already blocked will not unblock, you need to manually fix them and try test again.

Have reviewed your install and it's some sort of strange behavior with Joomla. Joomla isn't building the ACL tree properly as it should be auto-adding the Registered usergroup (you've renamed it, but that's ok), but for your custom usergroup it does not. However, for your other usergroups this works fine (for example Student works). Basically in short this is the only usergroup causing this issue so something about it is not setup correctly or some other extension is doing something to users with that usergroup.

You're using Joomla 2.5.6 which is out of date. I suggest updating Joomla to at least 2.5.9 and confirm if issue persists.