The symptoms of a defective cable vary greatly. I've had some where I could write to the drive during an OS install, but as soon as I would boot to the drive it would fail. Other times it wouldn't even power on an SSD. You can get a replacement cable from OWC. Any standard 2.5" SATA SSD should work just stay away from the cheap ones as they can be as slow as a hard drive (Crucial BX500 is one, Kingston is another). A Crucial MX500 works, an OWC SSD, and Samsung SSDs have been used by many.
You should be able to at least install macOS 10.11 El Capitan as it did not include a system firmware updater in the installer IIRC. If you were running High Sierra or Mojave, then you could try to see if they will install for you. The missing firmware partition error is because macOS needs to have a working internal drive for the system firmware updater. Maybe if you already had High Sierra or Mojave installed it wouldn't need to run the firmware updater with those versions of macOS. You can find links to other versions of macOS in this Apple article:
https://support.apple.com/en-us/HT201372