Personally I thought Bill just grabbed the first bag he could to look for a journal to put the mole print in. He didn't have a lot of time. So I think he randomly picked Dorothy's book.
I also felt that the players were so tired during the ball tossing game that they really didn't quite know what was going on, and that was the reason that Darwin accidentally missed his turn and no one else realized it.
As for why Anderson noticed it - I think they reviewed the entire tape, perhaps looking for someone to point suspicion to (especially if Bill is the Mole). By pointing out Darwin's error it seemed a deliberate attempt by Anderson to point out a "false mole."
Note that cameramen are always around and they probably informed Anderson of Bribs comments that he had deliberately touched the ball with his foot, and maybe they noticed Darwin's slip too.
A couple of the games have really been unfair, actually pretty much impossible to win - such as the wine carrying event (as proved when even Anderson couldn't pour without spilling a drop). I do wonder if they would have been allowed to wipe off the top of the bottle with their clothing or even suck off the top to eliminate drops. The gnome race was another that seemed impossible to accomplish, since there really wasn't enough time for all 3 people to do their jobs, even if no one dropped the gnome. I do wonder why they picked the bigger gnome - even if it was lighter in weight, it looked harder to carry than the smaller gnome. And having such a poor basket on the bike didn't seem fair either. Also penalties for disobeying the rules seem arbitrary too at times. Anyway, sometimes this show just doesn't seem entirely fair to the players.