I cannot make the search function work, sry.
How is it possible to make paratroopers move right after touch down? I thought there was supposed to be a 6 hr. delay? Or maybe this was changed while I was on break from the game?

This did not happen on an airfield, but at the line between two of my control points. It landed and started moving right away. No reorganisation symbol.
I have to say it was a pretty smart move that allowed him to attack my otherwise screened artillery from the back. To me it seems that the 6 hr delay only kicks in if you attack a control point. Up to now I even thought that wasn't possible to attack a line, but I think it's unbalanced if that kind of drop down somehow allows the paras to move.

Yes, I think that is what happened. He probably targeted my army. I must try it out.
However, I think it is wrong that it does not have a cooling down time.