Connect Appletv device directly to MacBook Air

no, the hdmi port in my macbook is a hdmi put that output to an external display much like the hdmi port in the appletv does, it's not an input port that support video signals being send to the mac

well

much of the stuff the appletv apps provide can often be accessed on a normal computer using a web browser or app

like netflix, appletv+ (on newer apps it would also work with the appletv app on the mac), amazon prime.....