I noticed some of the cards have At Midnight: instead of teror results. Do these automatically happen at midnight or only if a task is failed?

They automatically happen at midnight, as long as they're of the six adventure cards you have on the table. If they're in your trophy pile, they do not happen.

I have a question on this...

Considering that the game starts at midnight, if one of the six starting adventures has the "at midnight" effect, it is applied also before the first player has his turn or it is applied only when only the clock hand reaches again the XII after a complete circle?

I think it's should be applied also in the beginning, at least to spice up things...