Continuity mistake: When the two Cybermen meet the Dalek for the first time, they are exterminated. They fall directly in front of the Dalek, but when the camera angle changes to show the second Cyberman hitting the floor side on, you can see the Dalek in the background. The Cybermen both fell straight forward, so it shouldn't have been possible to see the Dalek when seeing the Cyberman side on in the next shot, as the Dalek is facing them both directly.
Continuity mistake: When Rose is hanging onto the lever and beginning to lose her grip, in close-ups of her face her body is facing the other wall of the room, perpendicular to the floor. In shots showing her and her hands on the lever, her body and head are facing the floor. (00:35:55)
Continuity mistake: After Pete tells the Doctor that Mickey went ahead because he wanted to find Rose, the Doctor asks him if he knows that Rose is his daughter from a parallel universe. The problem is that in Pete's last scene in "The Age of Steel", the Doctor saw Rose call Pete "Dad", leading to Pete departing in a hurry. (00:15:05)
Chosen answer: The Master knows that deep down, he deserves death for the crimes that he's committed throughout his life, and since he regards The Doctor as his arch-foe, he expects it to be at his hands. The fact that The Doctor is still willing to forgive him for all of his crimes hurts him more deeply than death would.
Captain Defenestrator