Revealing mistake: When the robotic Doctor first appears - and in many of the later scenes involving the jungle on Mechanus - it is very obviously not William Hartnell playing him, but a vaguely similar double. In addition, his lines have been dubbed by Hartnell and do not always match the movements of his mouth.
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Revealing mistake: In episode 1 when the Doctor, Ian, Barbara, and Vicky leave the TARDIS, wheels are visible which move the prop around. There is also no back to the prop.
Revealing mistake: In episode 1 when the Doctor is checking the controls you can see the studio lights to the top left corner where the set doesn't completely connect.
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 ★