Third-generation TCS runs Windows 2008, you should be able to upgrade to S6.2 if you continue to run the software on the hardware appliance, see the "Upgrading the Content Server Software" section of the TCS S6.2 Release Notes.
There is no direct upgrade path from TCS S6.x to S7.2. If you want to upgrade to S7.2, you'll need to install VMWare on the server and then deploy a Windows 2012 VM that the TCS software will run on, refer to the TCS S7.1 Virtual Machine Install Guide for Existing Customers for more information on this process.
A release key is needed if you're upgrading from one major version to another, ie: S6.2 to S7.2, however it is not needed if you're upgrading to a minor version, ie: S7.1 to S7.2.