If I run Black Sails, then one part of my deck turns into The Hold. Now I'm not sure that whether I can look it up anytime? And what about my opponent?

The Hold is an out-of-play area like your hand or Shadows - it's hidden from your opponent, but you can look at it any time you want.

I agree with ktom on this. Look at the wording on the agenda and Naval Reinforcements as far as getting a card out of the hold. The agenda says "choose a card…" and the plot says "reveal a character from…" Neither one says "search you hold for 'x'," so it would stand to reason that you would treat it the same as shadows cards or your hand, like ktom said.

Will The Hold be rebulit in the setup phase when my mulligan? Or only to shuffle the deck and remain The Hold unchanged?

Since the agenda redefines your play deck before you draw your setup hand, I'd say you leave the hold unchanged.

Khudzlin said:

Right. The agenda has you shuffle and split your deck before even drawing a setup hand (the same timing as when you choose your location for house of dreams), so if you want to mulligan, it's gonna be from the stunted deck that you had chosen to be your draw deck after shuffling.