Continuity mistake: When Astrid tells Mr. Copper that she's going down to Deck 31 to find the Doctor, she's using her left hand to fit a teleport bracelet on her right wrist. In a shot facing Mr. Copper immediately after as he wishes her good luck, her left arm is suddenly by her side. (00:50:55)
Revealing mistake: Max Capricorn is a cyborg, with his head being the only remaining part of his original body, wired up to a mobile life support system. If you look closely during his scenes, whenever he turns his head, the costume the actor is wearing to hide his body becomes apparent, as the wires and cables below Capricorn's head also move slightly.
Continuity mistake: When the Doctor interrupts Mr. Copper's incredibly inaccurate description of Earth society to ask him where he got his information from, in a shot facing Mr. Copper, the Doctor's raised hand has only one finger extended. In the next shot, facing the Doctor, he has two fingers extended, with the other two only loosely curled. Shortly after this, the Doctor scratches his temple with that same hand, only for it to be absent in the next shot facing Mr. Copper. (00:11:40)
Continuity mistake: After Rickston asks the Doctor how "Mr. and Mrs. Fatso" (Morvin and Foon) are going to get through a narrow gap, when the Doctor responds that they're going to make the gap bigger, he passes Rickston a piece of metal. The steel bar's appearance changes between shots. (00:27:45)
Continuity mistake: After Morvin falls to his death in the engine, when Foon tearfully demands to the Doctor that he bring him back, and he tells her he can't, Foon's hand is gripping the railing in one shot with the two of them, but missing in a shot facing the Doctor as he says, "I know. I'm sorry." (00:37:45)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When the Master says that the Doctor sealed the rift at the Medusa Cascade single-handed, in a wide shot, his right hand is resting on the edge of the table he's sitting on. The scene cuts to a close-up, and then back to a wide shot, where the Master's right hand is now in his lap. (00:10:30)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When Martha explains to Tom and Professor Docherty about how the Master could only use the TARDIS to travel between the present day and the end of the universe, and how he must have gone to Utopia, in a wide shot her right hand is resting on the table, but in a shot from above looking down on the captured Toclafane her right arm is by her side. (00:20:15)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When the Master says, "Out you come, little girl. Come and meet your Master," his arms go from lowered to raised between shots. (00:27:20)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When Martha walks up the Valiant's bridge, she passes Captain Jack and her family before stopping just short of the bottom of the stairs. In the next shot, she's still walking as she looks at the Doctor. (00:29:30)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When the Doctor is cradling the Master's body, in the close-up his right hand is curled up into a fist, but in the following wide shot is wrapped around the Master's shoulder. (00:41:45)
Last of the Time Lords (3) - S3-E13
Continuity mistake: When the Doctor and Martha react to discovering that Jack is the Face of Boe, the position of the Doctor's arms on the railing in front of them changes between shots. (00:45:05)
The Sound of Drums (2) - S3-E12
Continuity mistake: When Martha starts to phone her family, after the Doctor warns her not to tell them anything, as she snaps in response, "I'll do what I like!", in the first shot she's holding her phone away from her body, but in the second shot it's jumped to her ear. (00:14:25)
The Sound of Drums (2) - S3-E12
Continuity mistake: When Jack tells Martha they have to ditch her car after escaping from the Master's thugs, the position of her hands on the steering wheel changes between shots. (00:17:05)
The Sound of Drums (2) - S3-E12
Continuity mistake: After the Master calls Winters "the last President of America", in a shot from behind him and Lucy, he has his hands by his sides. In the next shot, facing them, the Master has his hands clasped behind him. (00:31:35)
The Sound of Drums (2) - S3-E12
Continuity mistake: When the Doctor stops and raises his hand while he, Martha and Jack are running through the Valiant because he senses the TARDIS, his fingers keep changing position between shots. (00:34:40)
The Sound of Drums (2) - S3-E12
Continuity mistake: When the Doctor fixes Jack's vortex manipulator so they can escape the silo, Martha and Jack are holding the lab door mostly closed against a horde of Futurekind. In the shot where the trio time-travels away, the Futurekind, some of whom had stuck their hands through the gap in the door, are missing. (00:00:40)
The Sound of Drums (2) - S3-E12
Continuity mistake: At the beginning of the Cabinet meeting, when the Master says, "Thank you. Thank you, one and all," in a wide shot, he turns his head to his right. In a close-up immediately afterwards, his head is facing straight ahead. (00:04:00)
The Sound of Drums (2) - S3-E12
Continuity mistake: During the Cabinet meeting, when the Master stands up as he says, "That wasn't funny," in a shot from behind he places both hands on the table as he rises. In the next shot, facing him, his right hand is suddenly by his chin. (00:04:10)
The Sound of Drums (2) - S3-E12
Continuity mistake: When the Doctor talks about how the Master was always sort of hypnotic, in a wide shot, Martha is sitting on the end of her couch, with her hands loosely folded, with her left hand on top. In the shot of her where she says she was going to vote for Harold Saxon, her right hand is on top. In the next wide shot, her hands have reverted to their original position. (00:12:05)
Utopia (1) - S3-E11
Continuity mistake: When everyone is being escorted into the silo after escaping the Futurekind, they all have mud splattered on their clothes - it's especially noticeable with the Doctor, Padra and Jack. Inside the silo, the mud has disappeared. (00:12:20)
Chosen answer: 1) When Stephen Moffat took over he ignored a lot of what had been developed before (there is not in-universe answer). 2) It would have killed Rose, so the Doctor absorbed the energy. His body regenerated before the energy could do a significant amount of damage that would prevent regeneration.