New PC install, Plex server no claim visible

You won’t be able to use the old server’s identity, so you need to create a new one. This then means, you will have to tell each and every Plex client to use the new server:
https://support.plex.tv/articles/customizing-the-mobile-apps-2/#reset
https://support.plex.tv/articles/customizing-the-apps/#reset
https://support.plex.tv/articles/customizing-plex-web/#resethome

Excellent. I recommend setting up the new server instance first.
Claim it into your plex.tv account: New Claiming Requirement for Windows
and set up one test library which you can point to a very small collection of video files. This is just to test principal function and transcoder working etc.

Once you have confirmed that everthing is working, you can either

If the backup of the plex data folder has been made while the server was still running, the primary database file is quite likely damaged. Restore one of the internal backups of this database file: https://support.plex.tv/articles/202485658-restore-a-database-backed-up-via-scheduled-tasks/