Other mistake: After the Doctor regenerates, there's a close-up of the Thirteenth Doctor's eye with the console room reflected in it. The reflection is the same orientation as a shot from behind her immediately prior, which should not be the case. Take note of the positions of the door and time rotor in both shots. (01:00:45)
Continuity mistake: When the Captain's arrival at the South Pole is first shown, walking up while Twelve is explaining to One that the frozen snow means something is very wrong with time and asking if either of them is a doctor, to which Twelve responds, "You trying to be funny?", the two Doctors are standing right next to each other. When the Captain's arrival is shown from his perspective, however, the Doctors are standing much further apart when Twelve says the line. (00:09:00)
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.