Is it possible to copy the remaining files of a (possibly corrupted) hard drive to an external hard drive? Let me explain. After trying to update my Mac Sierra (it was a 2013 Macbook Air model) to Mac High Sierra, the update failed ("MacOS could not be installed on your computer"), and I was unable to start up the laptop without encountering that error screen. I did find a "Startup Disk" option, but Macintosh HD wasn't there (there were no options at all). I tried using Disk Utility, found that Macintosh HD was unmounted, and attempted to use First Aid to repair any issues, but that failed. Then, I tried booting the computer in Safe Mode, using Terminal (my "r" key shorted out, so cmd+r on booting wasn't going to work - I had to copy and paste an "r" from the error log to place in Terminal). 

Upon checking, there was no available Time Machine device or directory I could use to save my computer, so that was out of the question. I intended to try to re-download and reinstall macOS, but then I found that my Macbook Air suddenly would not connect to my wi-fi, blocking me from that option indefinitely. From checking on Macintosh HD using Disk Utility, it appeared that my files still existed from the amount of storage taken up, but there was seemingly no way to access them or boot the Macbook Air from that HD.

My final hope is this: I have a 2TB external hard drive I could copy the existing files of the Macintosh HD to, and then access those files from a new Macbook Air (2018, 10.14.1, Mojave) entirely. Can this be done?
Assuming you haven't already solved this, each of the following options should work.  Also not sure if you went through with a new computer:


Start up in the recovery partition of an external bootable drive (especially if you have a USB recovery installer of Mac OS - highly recommended). Use disk utility to restore your external hard drive from the internal drive of the old MacBook Air. Once completed you should have an exact clone to hook up to your new MacBook Air or investigate a fresh install of the old one.
This is a slight variation, but if you haven't already got a USB recovery installer, you could partition your external drive, make the first partition into the recovery installer, and use the second partition to restore into a clone.
Start in disk utility on your 2018 MacBook Air (or a friend's Mac). Use a Thunderbolt 2 to 3 adapter and start your 2013 MacBook Air in Target disk mode (hold T on startup). If your hard drive doesn't mount, hopefully it appears in the device list for you to access.
Dismantle the old MacBook Air and place the hard drive into an external enclosure. Hopefully you have a friend with an enclosure you can borrow. If you buy a full upgrade kit, you may be able to salvage the old MacBook Air as well!