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.
Other mistake: The power output in the Doctor's shed reads "Megga Volts" instead of "megavolts."
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 ★