Putting cards into your hand while the top of library is revealed
Actually, I would argue that Anthony's opponent should not know where the revealed card ends up. When you manipulate cards in a hidden zone, what happens is not free or derived information. See rule 400.2 (emphasis mine)


  400.2. Public zones are zones in which all players can see the cards' faces, except for those cards that some rule or effect specifically allow to be face down. Graveyard, battlefield, stack, exile, ante, and command are public zones. Hidden zones are zones in which not all players can be expected to see the cards' faces. Library and hand are hidden zones, even if all the cards in one such zone happen to be revealed.


So I would think while resolving  Dig Through Time, and supposing X is on top of the library, the steps are:

(1) Look at the top seven cards of your library. [X is revealed at this point]

(2) Put two of them into hand and the rest on the bottom of the library in any order. [flip X over, perform this action, then reveal the new top card of the library]

Since the choices made for (2) above do not fall under the categories of free or derived info (library is still a hidden zone, see 400.2 above), I don't see why Anthony's opponent should know what happened to X.