All MacBook Air models from 2018 and later have USB-C ports, which can carry both video and audio to a monitor or HDTV. If the monitor also has a USB-C port, you can connect it to your MacBook Air using a USB-C cable without a special adapter.

I am using this monitor connected via USB C to an M1 MAC mini on Monterey and no, the keyboard Volume Control does not control the volume of the sound piped to the monitor. I think it would be fair to assume that on a MacBook the situation would be the same.
Monitors typically have their own volume control. So does this one. Hence, in general, this is a fair approach. However in some cases not having keyboard volume control for a monitor is inconvenient. I found there is software that can fix that.
I have «Sound Control» by «Static Z Software» [https://staticz.com/] running on my MACs. Among a lot of other stuff it has a per device option called "Force Keyboard Volume Control". This option overrides the direct sound coupling with the monitor and lets the keyboard volume keys control the volume. I found it useful and am happy with it.