I can not choose a startup disk

If you have the A1370 MBA, then there is no earlier model than 2010. Are you sure it's a MacBook Air? Nevertheless, if it's the 2010 model, then the only way to install Snow Leopard is with the Software Restore DVD that came with it when new. If it's from 2009, which you think it is, then it is not an 11" screen. The 2007, 2008, and 2009 models all have 13" screens. All three can boot from the Retail Snow Leopard DVD you can purchase from the Apple Store: Mac OS X 10.6 Snow Leopard - Apple Store (U.S.). The price is $19.99 plus tax. You will be sent physical media by mail after placing your order. After you install Snow Leopard you will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store. The MBAs do not have built-in optical drives, so you will need one to install Snow Leopard or another Mac with an optical drive from which you can make a bootable USB flash drive.

You need to first erase the drive before you can install macOS. But your computer cannot use Internet Recovery. If you erase the drive, then you cannot install El Capitan except from the Recovery HD. What you would do is this:

Install El Capitan or Later from Scratch

If possible, back up your files.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target drive (out-dented entry w/type and size info) from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the partition scheme to GUID.
6. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
7. Click on the Apply button, then wait for the Done button to activate and click on it.
8. Quit Disk Utility and return to the Utility Menu.
9. Select Install OS X and click on the Continue button.

However, what you posted from Disk Utility appears to be an existing system that is malfunctioning. It may be fixable by reinstalling macOS without first erasing the drive:

Reinstall El Capitan or Later Without Erasing Drive

Please be sure you back up, if possible.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the side list.
4. Click on the First Aid button in Disk Utility's toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Install OS X and click on the Continue button.

One the above procedures will work provided the drive hasn't failed and there is a functioning Recovery HD on the drive. However, if that is not the case, then your only remaining option is to boot the computer from a Snow Leopard DVD, erase the drive, install Snow Leopard, update Snow Leopard to 10.6.6, 10.6.7, or 10.6.8, and upgrade to El Capitan.

So, your model did not come with Restore disks when new? I take it that means it's a Late-2010 or Mid-2011 11" MBA. If it's the former, then it came with a chiclet-size flash drive with a version of Snow Leopard that was 10.6.4 or 10.6.5. The latter model is capable of doing an Internet Recovery. To find the model information select About This Mac from the Apple menu. If you have the 2011 model, then you can do this:

Internet/Network Recovery of El Capitan or Later on a Clean Disk

     If possible back up your files before proceeding.

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (out-dented entry w/type and size) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the partition scheme to GUID.
7. Set the Format type to APFS (SSDs, only if installing Mojave or Catalina ) or Mac OS Extended, (Journaled.)
8. Click on the Apply button, then click on the Done button when it activates.
9. Quit Disk Utility and return to the Utility Menu.
10. Select Install OS X and click on the Continue button.

This should install the most recent version of macOS that model can support. Both models can support up to High Sierra.

I am not sure but you can try this same procedure on the earlier model. I just don't remember if it supports Internet Recovery. In any case there is not harm in trying.

The computer must be connected to an Internet connection to work. Fast Wi-Fi or Ethernet. Wi-Fi needs to be selected and logged on for the recovery to work. You should be prompted to select your Wi-Fi network and login password during the startup process.

Let me get this straight. You booted the computer from the Recovery HD, then used Disk Utility to erase the drive. But you did not state that you then returned to the Utility Menu and selected Install OS X from the menu. There is no OS on the startup disk which is why you do not see one in Startup Disk preferences opened from the Recovery HD. Did I get this right? At this point I am not even sure that you installed a complete version of macOS (I assume you have an El Capitan installer.)