Corrected entry: When Spock takes Captain Pike out of the briefing room, Pike appears, via the viewscreen, on Talos IV in the very next shot. There is no possible way Spock could have gotten Pike to the transporter room that fast.
Corrected entry: At the end of part 1, Starfleet relieves Kirk of command and tells him and everyone else to stop viewing the Talosian transmissions during the Court Martial. At the beginning they say they can't stop the transmissions, go back into the courtroom, and watch them anyway. Why don't they just go to a different room, or cover the screen with a cloth?
Correction: Keep in mind the entire situation was being controlled by the Talosians. Even the commodor was an illusion created by them.
Correction: Not a mistake. Pike's appearance on the screen is of his younger, uninjured self - obviously one of the Talosians' illusions. So they "jumped the gun" a bit by showing him on the planet before his actual arrival, but this image is only for illustration to the people viewing, so it doesn't matter.