Ahoy, the fairpirate-- maybe I can help you out....
you'd have to boot up from the Recovery partition, to do that, shut your macbook air down, and press the power button, holding down Command/apple key and R together. if you see a language selection screen, it worked. pick one and go on. From there, use your trackpad/mouse (if you have a mouse, that is) and move the cursor (the arrow) to the top center of the screen and click. You should see OS X Tools. from there, you can choose to re-install the OS, but before that, run Disk First Aid from Disk Utility, and then re-install the OS. seeing as how you haven't yet installed Sierra, which is 10. 12.4, Option command R held down on startup gets you the OS that originally came with your Mac, or the closest version still available.
see what that does.
apart from Internet Recovery, which means plugging in an Ethernet cable into your MacBook Air, and holding down Command, Option and R together, after you do a shutdown of your mac. You'd see a spinning globe and a progress bar.
that's all I have for now
john b