Is anything keeping you from just reinstalling the system and mounting your home into it again (maybe the majority of your customisations live in /home too)? I feel that is a lot less of a hassle than copying files around.
In principle you should be able to restore your system by just copying all of the relevant files from the backup to their correct partitions - it can’t really get any worse if it doesn’t work.
For the future: A backup is only any good if you know how to restore it and tested that that actually works.
Regarding the permissions: If you do a cp fileA.txt fileB.txtfileB.txt will normally be owned by the creating user. So a sudo cp … will create the files as root.
I would personally use rsync with a few additional options, archive among them. This way the fs is restored exactly as it was. But that doesn’t make a whole lot of sense if the files weren’t copied that way too.
Nouveau is stable and runs, but don’t expect the best performance. The official NVIDIA driver is unstable, lacks proper wayland support but has decent performance. I’d go with anything but a NVIDIA GPU.