Those problems are due to corrupted data. Here's how to solve it:
- Make a backup of everything
- Get the full firmware on this page: https://www.playstation.com/en-us/support/system-updates/ps4/#newinstallation
It needs to be the full thing and not just an update. Should be around 900 MB in size
- Put the PS4UPDATE.PUP onto an USB drive inside a folder named UPDATE
- Put the USB drive into the PS4
- Go to safe mode (Hold Power Button in Rest Mode until the system beeps a second time)
- Select "Initialize PS4 (Reinstall System Software)"
- Restore your backup
You should then be able to use all of your hard drive again.
Official member of VGC's Nintendo family, approved by the one and only RolStoppable. I feel honored.