Related to the RDP issue:
The restore of the VM was finished successfully?
The restored VM is up and running?
Is there a NSG associated to the isolated vNet/Subnet or/and the restored VM? If "yes" there is an "allow rule for RDP TCP Port"?
Is a public IP associated with the VM?
Regarding the renaming issue:
It looks like a domain-joined computer can be renamed with Rename-Computer
if a DC is available.
To run your backup tests it would be helpful to have a restored AD in the isolated vNet/subnet as well.
There was a similar question a few days ago:
https://learn.microsoft.com/en-us/answers/questions/228266/how-to-restore-azure-vm-in-offline-mode.html
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(If the reply was helpful please don't forget to upvote and/or accept as answer, thank you)
Regards
Andreas Baumgarten