Your list of information could be any 11-inch MBAirs sold from 2011 to 2015 (5 different models)
the Cmd+Opt+R is, in fact, Internet Recovery, which will offer a version of OS X that is probably older than what you have installed now, and can be the original system that came installed when it was new. If the version is different, then you will have to erase the drive, losing all your own files. I think you need to try booting to the Recovery system first, and not Internet Recovery
Either way, before reinstalling your macOS, be sure to open Disk Utility (in the Utilities menu), and check to see that your boot drive is showing. Select that drive, and then run First Aid - a quick test of the drive.
If that is good, Quit Disk Utility, then go to the Apple menu, which should have the Startup Disk choice at the top. Is your internal drive show there? Select it by clicking on the drive icon. Try to restart now. If you still get the flashing folder, restart while holding Command+R
That should boot your MBAir to the System Recovery partition. If THAT is successful, you should try to Reinstall OS X, or macOS (whichever way that is showing). The system install files will download, and then will reinstall your system. This might take 30 minutes to an hour, so be patient. The speed will depend on the speed of your internet connection.
If NONE of this works, because your boot drive is not visible, then we will need more information about which specific model, out of 5 different ones, is the MBAir that you have. There is a label on the bottom cover that can help. Look for a model number, starting with A, such as A1465, also EMC number, such as EMC 2471