Revealing mistake: Perhaps criticizing the special effects is unfair, but the extremely obvious use of toy tanks and a rag-doll Sarah in certain scenes is one of the worst examples in the history of the show.
Revealing mistake: In the last part, the giant robot steps on a soldier. There are several things wrong with this scene. 1) You can see the matte halo around the robot's foot. 2) The robot does not leave a footprint. And 3), after the soldier gets squashed, his arm is visible waving over the robot's foot.
Revealing mistake: When Noah bangs his green arm down, in a close-up you can tell that the arm has green bubble wrap around it. (00:02:30)
Revealing mistake: When Vira emerges from the hibernation chamber, the Styrofoam that forms the pallet squeaks.
Revealing mistake: As the cast move through the transom of the space station, observe the starscape through the lower windows. At certain points you can see a thin strip of light where the black curtain used for the stellar backdrop meets the studio floor.
Genesis of the Daleks - S12-E4
Revealing mistake: During the Doctor's early conversation with the Time Lord, there is fog swirling around them. When the Doctor agrees to take on the mission, it suddenly appears very concentrated around the tip of the rocks behind them, as if generated from a source behind the rocks. Normal fog would not do this, and it's clear that dry ice was used as the fog effect (DVD commentary confirms this). (00:02:45)
Genesis of the Daleks - S12-E4
Revealing mistake: When Ronson is killed by The Daleks, you can see Ronson reacting a little too soon to the Dalek laser. (00:14:40)
Genesis of the Daleks - S12-E4
Revealing mistake: In episode 6, when the Thals detonate the charges, smoke is seen coming through the door before the detonator is pressed down. (00:19:40)
Genesis of the Daleks - S12-E4
Revealing mistake: Whenever there's a close-up of Davros, it's very easy to see that there's a gap around his mouth where the mask should blend in with the actor's face, but doesn't. This mistake was rectified, incidentally, in Davros' next appearance in "Destiny of the Daleks."
Revenge of the Cybermen - S12-E5
Revealing mistake: At the climax of episode four, the Beacon plunges close to Voga and orbits very fast before the Doctor manages to break free. The planetscape that appears on the screen is quite obviously a revolving drum with bits of clay on it.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★