2013 iMac as display for MacBook Air (M1 2020) - what cable?

You could make a Thunderbolt data connection using a $50 Apple Thunderbolt 3-to-2 adapter, and a $30+ Thunderbolt 1/2 cable … but there's no point. Your M1 MacBook Air is too new to use any iMac as a Target Display. It would be $80+ down the drain that you could have put towards purchasing a standalone monitor.

See the bad news here: Use your iMac as a display with target display mode - Apple Support

Also, your iMac is too old to be an AirPlay Receiver for AirPlay to Mac. (There is third-party software that might allow it to act as a sort of "AirPlay Receiver", but AirPlay is not as good as a connection to a real monitor.)

The 2020 MacBook Air does not support using the 2013 iMac as a display.

please carefully read > Use your iMac as a display with target display mode - Apple Support

If you want to use a second display with the MacBook Air.

see > Use an external display with your MacBook Air - Apple Support

No, because Target Display mode does not work on the 2020. You can not use the iMac as a monitor.

Yes any of those monitors will work with the 2010 MacBook Air.

You can use any one of the following connections:

a USB-C to HDMI adapter and HDMI cable.

a USB-C to HDMI adapter cable.

a USB-C to DisplayPort cable.

Sorry the 2010 was a typo...

Yes you need to use a USB-C adapter or adapter cable.

You're welcome.