How can I move my VMs to the new server? I've been doing backups on secondary SSD drive on the old server and installed that disk in the new server, however I cannot find my backups. How can I access them?
Really depends on what storage you created your backup disk with.
In general:
1.) Mount your filesystem (depends on filesystem/storage hiw to do that)
2.) add the existing filesystem as a new storage at datacenter -> storage -> Add (again depends on what filesystem was previously used) and make sure VZDump Backups are a selected content type
3.) In case you use a directory storage make sure to manually set the "is_mountpoint" option via CLI
4.) Select your newly added storage and restore guests from its "Backups" tab
Is it possible to restore a VM from a backup created on a different version of Proxmox? In my case, backups were made in Proxmox 5.4-13, and I would like to restore them in the current version of Proxmox 8.0
Is it possible to restore a VM from a backup created on a different version of Proxmox? In my case, backups were made in Proxmox 5.4-13, and I would like to restore them in the current version of Proxmox 8.0
Not sure about that big jump, but in general restoring backups on a newer PVE works.
But read the known issues in the roadmap for PVE6, 7 and 8. Old LXCs might not run anymore without workarounds as cgroup v1 support got dropped etc.