Do scenario specify where excatly clue cards (A,B,C) should be placed? Is it the same room everytime, or this can vary?

Do keeper place all of them on the board, and what happens if players find clue 2x before clue 1x? x mean A, B, or C – depends on keeper’s choice.

From my understanding the Keeper's Guide will instruct where all the clues go and (possibly) all the other cards as well. Also, I think clue #1 is the clue that informs of the Investigators' objective, so hopefully that would be the last clue found.

Additionally, as far as I'm aware, all the rooms that contain clues are locked, and the "key" (whatever form it might take) can only be found at the location of the previous clue.

All the cards are placed on the board before the game starts, depending on how the Keeper answers his questions during setup the cards will be placed in different rooms, and from the scenarios I've run so far it looks like the clues can only be found it order, though I can easily see being able to design one where this is not required.

After looking through all the cards, there is one item card ("Startling Evidence") that requires the Keeper reveal which room the next clue is in. So I suppose the clues must follow some kind of linear order.

Yes, the clues are found in order with clue 1 being found last.