Trivia: As Kirk, Spock, and Uhura head to the shuttle bay for their mission to Kronos to capture Khan, Sulu orders crew to prepare the vessel that was confiscated in the Mudd incident. This is a reference to Harry Mudd, a roguish character who appeared in the original Star Trek series in the episodes titled,"I, Mudd" and "Mudd's Women." Harry Mudd, played by Roger C. Carmel, was the only non-regular character to appear in more than one episode on the original Star Trek series. Carmel was slated to reprise the role in Star Trek: TNG, but he died before the episode could be filmed. (00:44:25)
Trivia: The library that is bombed is called the "Kelvin Memorial Archive," a reference to the destruction of the U.S.S. Kelvin seen in the first film. (00:18:05)
Trivia: During the conversation between Admiral Marcus and Kirk, right after Marcus shows up in his new ship, look at the display over Kirk's left shoulder. It's out of focus, but looks like 1701 and the name GENE. In other views of the display, it looks like the letters C and O. Not sure if it was deliberate, but interesting if it was an homage to Gene Rodenberry.
Trivia: Admiral Marcus looks at several model ships when explaining Section 31 to Kirk and Spock. One of them is the Phoenix from "Star Trek: First Contact" and one of them is the Enterprise NX-01 from "Star Trek: Enterprise." (00:21:35 - 00:31:25)
Trivia: Admiral Marcus is looking at models of various air and spacecraft that were very important to history. These include (in approximate order) the Wright flier, an early propeller plane, a V2, what appears to be a Bell X-1A, sputnik, a Mercury space capsule, a space shuttle, the upcoming Space Launch System (SLS), an unknown spacecraft (which might be also referenced in Star Trek the Motion Picture), Zephram Cochram's ship the Phoenix (and presumably the version immediately after that), NX-01, the USS Kelvin, and the Vengeance.
Chosen answer: The explosion of the Moon Praxis in the original Universe was due to extensive over mining and energy production. In the first movie that took place in the alternate reality, an entire Klingon armada was destroyed by the Narada. It is logical to assume that the Klingons began to over-mine the moon in order to obtain the resources necessary to replace so many lost ships, causing the moon to explode several decades before it happened in the Prime timeline.