In rescuing Rudge, Steed inadvertently turns on the walkie talkie. Those devices had 'push to talk' switches; you had to deliberately hold down the switch in order to transmit. They were specifically designed to preclude accidental transmission.
The mouth and eyes of the ventriloquist doll move, even when Henry is not manipulating the controls on his left wrist.
When Mrs. Peel follows Ms. Merryweather to the Throgbottoms Class, You can see a cameraman's boot in the bottom left.