There wouldn't be any issues upgrading from TC3 to 7. Just make sure you read all the release notes of all the versions between where you're at and want to go to so you'll know if there might be any issues or changes to expect. One thing comes to mind is in TC6 and it's minor releases, they changed how layouts, selfview, and dual displays behave and are configured.
If the codec has a valid service contract, you'll just need to get a release key for the software version you want to install. I typically add the release key to the codec, then upload the software and start the upgrade. Entering the release key will prompt to restart the codec, however if you wait, the software upgrade will restart it anyway, just saves a little bit of time not waiting on an extra restart.
Note, from TC6.1 onward, release keys are no longer needed for software upgrades, just a valid service contract attached to your Cisco login to download the software.
If you don't have an active service contract, you should be able to get an upgrade through the TAC as per the "Customers Without Service Contracts" section in following security advisory - cisco-sa-20140605-openssl.
Are you getting this error “Installer User Interface Mode Not Supported. The installer cannot run in this UI mode. To specify the interface mode, use the -i command-line option, followed by the UI mode identifier. The value UI mode identifiers...
The below trick might come handy when you have to add a new node to a cluster but you don't have or is unsure of the security password for the publisher. This procedure has been around for ages.
1) Login into the CLI of the Publisher.