In TNG, the lower the level of something the higher in intensity it was (a Level One Diagnostic was more intensive than a Level Four). In the 32nd century, the opposite seemed to be the case, as Reno talks about a Level-10 diagnostic in SFA: “Rubincon”. SNW seems to be taking the latter approach as well. This episode is not the first time we’ve seen people turned into puppets - in LD: “Kayshon, His Eyes Open”, Cerritos security chief Kayshon was turned into an (albeit inanimate) puppet by an energy beam. Fans might also remember the Angel episode “Smile Time”, a parody sequence in Stargate SG-1’s “200” and an imaginary sequence in Community’s “Advanced Documentary Filmmaking”.

The stardate is 3019.4. Enterprise is patrolling Rizonian space (first mention) on her way to Vulcan. Space piracy is indeed extant in Star Trek, with various species and factions engaging in piratical or privateering activities, notably the Orions (TAS: “The Pirates of Orion”), the Ferengi (TNG: “Rascals”) and the Breen (DS9: “To the Death”).

Spock and Scotty discuss longer-range transportation. Spock says 27,000 km is 2,000 km further than any transport achieved. By the 24th century, transporter ranges on the Galaxy-class could reach up to 40,000 km (TNG Tech Manual). To be honest, 27,000 km seems a bit short, since Earth geosynchronous orbit is 35,786 km. This might be a “Sci-Fi Writers Have No Sense of Scale” trope issue.

That Scotty is working on increasing transport ranges references Star Trek (2009), where the Kelvin timeline Scotty is working on transwarp transportation to allow transmission across interstellar distances. Prime Spock helped him with this, giving him the necessary equations his Prime counterpart had already solved. 
 While chanting the mantra, “Logic is the rhythm of your katra,” (katra being roughly the Vulcan equivalent of the soul, ST III), Spock hears echoes from SNW: “The Griffin Incident”.

Kal Rekk, a day of “atonement, solitude and silence” was first mentioned in VOY: “Meld”.

T’Pring’s suspicion of “hijinks” refers to her two previous trips to Enterprise, one which resulted in a body swap (SNW: “Spock Amok”) and another which had Spock turned physically human (SNW: “Charades”). Specifically, in “Spock Amok” while in Spock’s body she says she does not like hijinks.

When Spock turns to see the puppet T’Pring, the music echoes bits of the soundtrack to TOS: “Amok Time”, the episode that introduced her and Vulcan.

April is actually less nonplussed about the situation that would be expected. But then again, as Janeway said in VOY: “Deadlock”, “We’re Starfleet officers. Weird is part of the job.”

La’An has been concerned since SNW: “Four and a Half Vulcans” that her Augment heritage is asserting itself.

Orion interceptors first appeared in ENT: “Borderland” and are playable Tier 6 ships in Star Trek Online.

This is the second time in the series that someone has captured Enterprise, the previous time being Captain Angel in SNW: “The Serene Squall”. To paraphrase Oscar Wilde: to lose a ship once may be regarded as a misfortune; to lose it twice looks like carelessness.

While the stereotype of the submissive Orion slave girl is prevalent in popular culture, it is true that Orion women control the males via the use of pheromones (ENT: “Bound”). Another matriarchal society was seen in TNG: “Angel One”.

Spock “touching hands” with Pike while separated through glass and Pike saying, “The needs of the many…” echoes Spock and Kirk’s scene at the end of ST II when Spock dies. The full phrase - a Vulcan aphorism - is, “The needs of the many outweigh the needs of the few, or the one.” Kirk, naturally, disagrees, sacrificing his ship and career to save Spock in ST III because “The needs of the one outweighed the needs of the many.”

Wait, how did Pike get back into the ship (and that room) if none of them can operate doors and they couldn’t even get out of the room? This is answered at the end of the episode.

Pike, La’An and Una in Vulcan robes is the first time the classic totem pole trench trope has been used in Star Trek.

When T’Pring rescues Spock, we can see a Romulan ale bottle behind the counter, although the liquid inside does not appear to be Romulan ale blue.

While the phrase “my mind to your mind, my thoughts to your thoughts,” is forever associated with mind melds, it was not actually used in TOS. The phrase debuted in TAS: “The Infinite Vulcan”, said by Spock’s giant clone.

And to round off a trilogy of tropes, we have the “All Just a Dream” trope, which was used to probably better effect in DS9: “Far Beyond the Stars”, although the dream appears to have had some therapeutic effect for Spock.