Visible crew/equipment: (Ep. 2) When The Doctor, Jo, and Dr. Tyler are taken into Omega's palace, in the first shot of the multiple passageways (right before Jo says that it looks like Aladdin's cave) the curved chalk actor's mark is visible on the floor, then we see it again in three following shots. Note that when they're all escorted further inside, just as Dr. Tyler comments about the matter/anti-matter not making sense, Jon Pertwee looks down, presumably to make sure he hits his mark.
Visible crew/equipment: You can see the reflections of the camera crew in the monitor used to display the First Doctor (William Hartnell). (This is especially noticeable in episodes 2 and 4).
Visible crew/equipment: (Part 3) After Omega has revealed himself to The Third Doctor, Jo, and Dr. Tyler, when he begins to speak privately to The Doctor the camera casts a shadow on Omega's body. Then after Omega turns around the boom mic dips into view at the top of the screen. And when Omega starts walking forward, the shadow of equipment (presumably the boom and mic) is moving on the far wall, near the lower left of the screen. (00:03:10)
Visible crew/equipment: When the group are thrown into a dungeon in Omega's palace, you can see tape in the corners which is holding the set together.
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 ★