According to Star Trek canon, Pavel Chekov was born in 2245. This movie takes place 10 years after the USS Voyager returned to Earth from the Delta Quadrant (Voyager returned in 2378 according to Star Trek canon, so this would make it 2388), so this would place Chekov's age at approximately 143 years old.
Captain Alvarez is the captain of the USS Archer which is named after commander Jonathan Archer who was captain of the first interstellar ship enterprise.
The quote that Lexxa Singh recites is from the poem Invictus by William Ernest Henley
According to Star Trek Canon (as well as Star Trek: Voyager (1995) and Star Trek VI: The Undiscovered Country (1991)), Tuvok was born in the year 2264, which would make him 124 years old at the time setting of this film (2388).
No Stardate is given at any point in the episode, merely that it is set in the year 2388. Since this is set in the original Star Trek time line, and 10 years after Star Trek: Voyager (1995), the appropriate Stardate would be somewhere in between Stardates 65000 and 66000.
Edward Furlong is best known for his role in Terminator 2: Judgment Day. In this film he becomes one of several Terminator veterans to appear in a Star Trek project. The original film featured Paul Winfield from Star Trek II: The Wrath of Khan. The second featured Jeanette Goldstein from Star Trek Generations. That film also featured Thomas Dekker, who played John Connor in The Sarah Connor Chronicles. In Terminator Salvation, John's father Kyle Reese is played by Anton Yelchin, who played Chekov in Star Trek Into Darkness.