Audio problem: In episode 3, when the dog is barking at Sam's house and is abruptly silenced (apparently by an Auton), the sound is obviously not a dog barking. It sounds a bit like a man pretending to be a dog, and going "woof woof" not very convincingly.
Audio problem: When the Doctor arrives at UNIT HQ in his stolen car, he speaks to a guard. We see close ups of the guard twice...but watch the brief clip of the Doctor (Jon Pertwee) between the two and you can just see that his lips aren't moving even remotely in synch with the words. (In fact, it is just a repeat of the previous shot of the Doctor, with new lines dubbed in).
Revealing mistake: When the Doctor falls out of the TARDIS, near the beginning, he nearly manages to pull off the entire TARDIS door.
Audio problem: The music accompanying the closing credits fades down part way through (at a different point in each of the four episodes) and simultaneously fades up at a different point, the net result being a rather disjointed-sounding edit.
Other mistake: I know that regeneration causes the Doctor to change his appearance, but I didn't know it can produce a tattoo visible on the Doctor's right arm in episode 2.
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 ★