I've set up a case with different security rights for different users to be used as a template. What I've noticed though is that when I use the template, it's not copying over the security settings that were originally set up for the template. Is this how it's supposed to work, or is there a way to copy over the same security settings into the new case the template is being used for?
Security is not transferable for database templates, this is how it should be working. The only way to "transfer" security is via Replication. This would not be a bad solution if an administrator is creating the databases, but not an ideal solution if the users are creating the databases.

Can security be done on Concordance wide level or is it on the database level? That is, can I set security for a user so that any database he uses has those security settings.
Security is set in Concordance on a database level NOT on a system wide basis.

Can security be turned off if you decide not to use it anymore?
Simply uncheck "Enable security" to turn off current security settings.

The way I understand the documentation, if I do not select Login Required, that the system will try to determine the userid and log them in automatically as long as a password is not required. Is this correct?
Concordance will take the system login (ie- "emartin") and compare against users in the security file. If it finds a match, it will use the assigned rights. Default rights are given to any login that is not matched. In general, the Default rights are assigned with the greatest number of users in mind or for "guest users" that should not have access to editing functionality.