If your MacBook Air's internal drive is not showing up as a bootable option in the "Choose Startup Disk" menu, and you're unable to install macOS Sierra through the recovery mode, you can try the following steps to create a bootable USB drive using the Yosemite disk image:
Please note that creating a bootable USB drive will require another computer with a working macOS system. If you don't have access to a working Mac, you can try borrowing one from a friend or visiting an Apple Store or authorized service provider for assistance.
Here's what you can do:
1. Obtain a USB drive: Get a USB drive with at least 8GB of storage capacity. Make sure it's empty, as the process will erase all existing data.
2. Download the macOS Sierra installer: Although you mentioned having a disk image of macOS Yosemite, it's recommended to use the installer for the same version of macOS that was originally installed on your MacBook Air. If possible, download the macOS Sierra installer on the working Mac from the Mac App Store.
3. Prepare the USB drive: Connect the USB drive to the working Mac. Launch the Disk Utility application (you can find it in the Applications/Utilities folder).
4. Erase the USB drive: In Disk Utility, select the USB drive from the left sidebar. Click on the "Erase" tab and choose a suitable name, format (Mac OS Extended Journaled), and scheme (GUID Partition Map). Then click "Erase" to format the drive.
5. Create a bootable USB drive: Once the USB drive is formatted, open Terminal (you can find it in the Applications/Utilities folder) and run the following command:
sudo /Applications/Install\ macOS\ Sierra.app/Contents/Resources/createinstallmedia --volume /Volumes/[USB Drive Name] --applicationpath /Applications/Install\ macOS\ Sierra.app
Replace `[USB Drive Name]` with the actual name you assigned to the USB drive in Disk Utility.
This command creates a bootable macOS Sierra installer on the USB drive. It may take some time to complete.
6. Eject the USB drive: Once the process is finished, you can eject the USB drive using Disk Utility or by right-clicking on the drive icon on the desktop and selecting "Eject."
Now that you have a bootable USB drive with macOS Sierra, you can try booting your MacBook Air from it by following these steps:
1. Turn off your MacBook Air.
2. Connect the bootable USB drive to one of the USB ports on your MacBook Air.
3. Turn on your MacBook Air while holding down the "Option" key.
4. You should see a list of bootable drives. Select the USB drive with macOS Sierra on it.
5. The installation process should start. Follow the on-screen instructions to install macOS Sierra on your MacBook Air.
If you encounter any errors during the installation process, it's possible that there are underlying hardware issues with your MacBook Air or compatibility problems between macOS Sierra and your specific model. In such cases, it's advisable to reach out to Apple Support or visit an authorized service provider for further assistance.