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)
Utopia (1) - S3-E11
Continuity mistake: When the Master takes the disc with the location of Utopia out of the computer, the exact spot he's holding it changes between shots. (00:40:55)
Continuity mistake: When Sally and Larry are hugging each other as the TARDIS dematerializes around them, as the cellar with the Angels becomes more visible, Sally lets go of the console with both hands and clasps them together. A couple of shots later, her left hand is still touching the console. (00:37:55)
Continuity mistake: When Sally enters the DVD store looking for Larry, she encounters the store's owner, Banto, who is sitting behind the front counter watching a movie. He holds up his finger to indicate her to wait briefly because of the point in the movie he's at. In shots facing him, his hand is quite close to his mouth and chin, but in a shot facing Sally, his hand is considerably further away from his head. (00:13:10)
Continuity mistake: When Sally picks up the remote to pause the recording of the Doctor after getting unnerved by him seeming to hear her, and when she rests it against her forehead after Larry comes back into the room, it switches between her hands a few times. (00:16:40)
Continuity mistake: After Sally leaves the garage, Billy turns his head and sees the Weeping Angels around the TARDIS, and his facial expression changes from a smile to a confused frown. In a close-up of Billy after the shot of the Angels, his head is turning and his expression changes a second time. (00:20:40)
Continuity mistake: When Sally is holding the picture of Billy and his wife, the position of her right index finger on the frame changes between shots. (00:24:35)
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.