Vulcan has always been a desert planet with an Earth-like atmosphere. In this episode it has a green atmosphere in some shots for some reason.
Tilly is appointed Executive Officer over more senior officers, without an acting promotion, and without complaint or concern of any of the more senior officers. This defies the basic and universal concept of military rank. The purpose of rank is to denote seniority in terms of authority and responsibility. The seniority of the Executive Officer is second only to the captain, and rank held by the Executive Officer must reflect this.
The planet Discovery visits is named Ni'Var for the Reunification of Romulus and Vulcan after the disaster. On Star Trek Enterprise, Season 1, Episode 12, "Shadows of P'Jem", the Vulcan Ship Ni'Var arrives to take T'Pol back to Vulcan for questioning, two centuries before the disaster happened.
Admiral Vance informs Burnham and Saru that the planet Vulcan is now known as Ni'Var. Apparently, at no point in her year of exploring the galaxy before Discovery caught up to her, did Burnham inquire about the state of the world she grew up on.
The footage that Burnham watches of Leonard Nimoy as an older Spock, taken from Unification II (1991), is from a private conversation between Spock and Jean-Luc Picard in a cave on Romulus. There was no one recording that conversation.
Burnham and Tilly deduce that because there was a delay of ships going offline of a fraction of a microsecond, the Burn must have had an origin. There are innumerable other variables that could have affected when each ship destroyed in the Burn went offline (speed of travel, type of warp engines/core, modifications made to the warp engines/core, power usage, etc.), but those are never brought up or explored.
Burnham and Tilly agree that Burnham is not in Admiral Vance's good graces and she should not be the one to present their findings regarding the Burn to him. The very next scene is Burnham (with Saru) presenting their findings to Admiral Vance with no mention of a change of mind on anyone's part.
Saru credits the Vulcans with coining the phrase, "The needs of the many outweigh the needs of the few," a phrase Spock used in Star Trek II: The Wrath of Khan (1982). The phrase, or a slight variation thereof, has been used throughout Earth's history in many cultures and philosophies, long before the time period in which humans are said to have first interacted with Vulcans.