I have created a lot of airfighter and when the enemy attacked me all of my airforce disappearfrom battle field even my aircraft carrier with airfighter... Is it bug or by design to make airforce useless in the game?

Airforce is extremely useful in HK, but they can't participate in fighting if they got stuck inside the battlefield. There's also, iirc, an UI issue when trying to make an Air Strike into an ongoing battle.

And they have to be not in patrol the same turn, if I remember well.

Yeah I think it is pretty ridiculous that a carrier can't deploy aircrafts when in battle. But if they are on the outskirts of the fight you can have them attack by backing out of the fight and going to the regular map and then click on the airplanes that way. But I think a carrier in battle should still be able to use planes.

I agree, but I think it should come at the cost of some initial damage done to the planes, if they have to scramble mid-combat.