The U.S.S. Archon must have been launched right around the time Star Trek: Enterprise takes place--around the 2150s, in order to have been lost "more than 100 years ago" at the time of this episode in 2266. Strange that Starfleet waited so long to investigate...
Kirk says halfway through the episode that Scotty is now in command. He would have been in command before then.
Trivia: This is the first time we hear of the Prime Directive
When Kirk goes to destroy Landru, Landru deactivates his phaser. Yet a few moments later, Kirk draws it to fire on a Lawgiver approaching him, despite already knowing it has been deactivated.
Kirk states that they are at the planet to find some trace of the starship Archon that disappeared 100 years ago. Later on, once they have get their missing man back, Kirk refuses to leave because he says, "we still haven't found out what happened to the Archon". If the Archon disappeared 100 years ago, what exactly were they hoping to find at this time? After 100 years, all the crew would likely be dead (unless it was a Vulcan ship, which it wasn't). Also, since Landru pulled the Archon into the atmosphere, it would have burned up so there would be no trace of the ship either.
Like in so many episodes, Kirk and company make huge generalizations about an entire planet based on a single experience. When Kirk and company beam down to the planet, they are there for a grand total of about three minutes before Kirk looks at all the "absorbed" people and says "If everyone on this planet is like this...". That's a pretty big assumption. Sort of like landing on Earth, seeing five drunk people and then saying "If everyone on this planet is like this!...".
You can briefly see a trail of burnt gunpowder leading up to one of the panels that exploded when the Landru computer blew up - the smudge disappears in subsequent shots.
Kirk and Spock approach Landru's chamber via a long hallway leading straight up to the doors. But when they enter the chamber, the audience can see the open doors behind them and there's a solid blue wall there instead of the long hallway.
There's a flip-shot of Kirk with his part on the wrong side right after McCoy returns to the dungeon.
When Landru appears in the dungeon, one of the security guard actors misses his cue and covers his ears a couple of seconds before Landru starts his sonic bombardment and everyone else reacts.
When the landing party goes upstairs at Reger's house, the windows are pitch black in the far-shots, but become transparent in the close-ups with Kirk.
As Festival begins, the landing party runs for cover. Watch and you can see a big thrown rock bounce off the head of one of the security guards. He doesn't even seem to notice.
Reger's light panel initially radiates light in all directions with no shadow, but when he places it on a shelf it casts a shadow.
At the beginning of the episode the Lawgivers convert Sulu just by touching him with a staff and it is permanent even at a long distance up in orbit. Why don't they do this later with Kirk's party? Instead, they haul them off to some big absorption chamber.
Kirk and his team ignore the Prime Directive by beaming down in plain sight before they know anything about the planet or its inhabitants. In other episodes they always beam down secretly.
Landru claims there is no disease on the planet, but later McCoy gives Reger's daughter a shot and the locals act like they know what he's doing.
Landru Computer: I am Landru. I am he. All that he was I am, his experience, his knowledge.
Kirk: But not his wisdom. He may have programmed you, but he could not have given you a soul. You are a machine.
Landru: Your statement is irrelevant.
(to the Lawgivers)
Kirk: You can get rid of those robes. If I were you, I'd look for another job.
Spock: Predictably metaphysical. I prefer the concrete, the graspable, the provable.
Kirk: You would make a splendid computer, Mr. Spock.
Spock: That is very kind of you, Captain.
Spock: How often mankind has wished for a world as peaceful and secure as the one Landru provided.
Kirk: Yes. And we never got it. Just lucky, I guess.
Kirk: Without freedom of choice there is no creativity.
Kirk: It's time you learned that freedom is never a gift. It has to be earned.
Lindstrom: Just wanted to say goodbye, Captain.
Kirk: How's it going?
Lindstrom: Couldn't be better. Already this morning, we've had six domestic quarrels, and two genuine knock-down drag-outs. It may not be paradise, but it's certainly human.
The concept of "Festival" was never explained within the show. It's certainly not necessary to explain everything, and in fact one of Star Trek's strengths was its often tacit assumption that the viewers could figure some things out for themselves. However, one wonders how "Festival" advanced the plot -- why include it at all? There was certainly ample evidence that something unusual was going on prior to this. In his adaptation of the episode, James Blish did provide an explanation. His explanation: Landru's idyllic society lacked population control, so Festival was conceived as an answer, the sort of solution a machine without soul or wisdom might create.
The U.S.S. Archon was of the Daedalus class.