Can I switch to proprietary drivers with a GPU and a CPU with integrated graphics?

That’s probably a remnant from a Windows installation that happened at some stage. Or something, I’m unsure.

What I am sure about, however, is that I don’t really know what’s going on here. There are a lot of partitions there, and while I’m not saying it’s wrong, I am saying that it’s confusing the living daylights out of me.

Assuming you don’t have any data you want to keep on the SSD, this would be my suggestion, though:

  1. Disconnect the HDD completely; ensure only the SSD is connected.
  2. Boot into the BIOS/UEFI and make sure that securee boot, fast boot and any RAID is disabled.
  3. Bot the computer into a live environment, format the SSD with the ext4 file system; and
  4. reboot into the newly installed OS to make sure everything is good.
  5. Turn off, connect the HDD and go with this:

This should go without saying, but if there is important data on the SSD, first MAKE PROPER BACKUPS because this will format the disk, thus removing all said data!