I'm not sure in which order i resolve Rook's ability and cards like "through the gates" or "crisis of identity (Bonus: What about research cards).

What order do i resolve those effects?
As i understand it:

1. Start search, trigger research effects, as their "when" triggers during the search, before drawing any.

2. Draw the chosen normal card (could be a weakness as well)

3. Weakness's Revelation triggers. Would "through the gates" discard the topmost card of those that i'm searching? So if i chose the topmost card at step 2 i'd discard the next one.

4. Shuffle your deck from rook

I think that's actually correct, except that the revelation effects from the weakness you draw should occur after Rook's ability has completely resolved. Mandy's weakness specifically interrupts the search, which creates a nested sequence inside Rook's ability, but most weaknesses don't interrupt the ability that draws them. Likewise research abilities create nested sequences because of their "when" usage, as you say.

Rules about weakness revelations say: "When a weakness card enters an investigator's hand, that investigator must immediately resolve all revelation abilities on the card as if it were just drawn." In my example above that's step 3 (before shuffling).

Now i see, the revelation has no "when/ if/ after" trigger so it can't create nested sequences? So it waits until rook is done. Thanks!