FAQ for
Star Trek Beyond (2016) More at IMDbPro »

The movie itself does not provide an answer to what happened to Carol Marcus (Alice Eve). She was still aboard the Enterprise at the end of Star Trek Into Darkness when the ship left for its 5-year mission, stating that she was happy to be part of the "Enterprise family". Star Trek Beyond takes place 2.5 years after Star Trek Into Darkness, but Carol Marcus is nowhere to be seen in the movie, nor referenced.

According to behind-the-scenes stories, Carol Marcus was scheduled to re-appear on Star Trek Beyond in the early script stages, based on ideas from producer J.J. Abrams. The movie was initially conceived as a direct sequel to Into Darkness, and would follow-up on some of the plot elements introduced in that movie. However, during pre-production, writer and initial director Roberto Orci left, and writing duties were taken over by Doug Jung and Simon Pegg. When this new team chose to take the story in a totally different direction, most of the ties to Into Darkness were also removed. Carol Marcus' presence was probably omitted because they could not fit her into the new story. According to Eve herself, there is still a good possibility that her character will makes an appearance in future Star trek movies. (Sources: http://www.bustle.com/articles/174622-why-isnt-carol-marcus-in-star-trek-beyond-alice-eves-doctor-was-left-behind, http://www.bustle.com/articles/106628-will-carol-marcus-be-in-star-trek-beyond-alice-eve-talks-returning-to-the-franchise).

So in story terms, it seems most likely that Carol Marcus has left the Enterprise for unknown reasons prior to the events of Beyond.

The adversary Krall is ultimately identified as a deformed Balthazer M. Edison, captain of the U.S.S. Franklin, a ship that, according to Scotty, disappeared without a trace in 2164, shortly after the foundation of the Federation of Planets.

Krall/Edison is also identified as a former M.A.C.O. (Starfleet Marines), and mentions fighting during the Xindi War (2153-2154) as well as the Romulan-Earth Wars (2156-2160). Especially the mentioning of the Xindi War is interesting, because this places him directly on board the Enterprise NX-01, during the events of the third season of Enterprise. He later states that he became desillusioned when the Federation was founded, and there was no apparent need for soldiers anymore. He received command of the U.S.S. Franklin in recognition of his services for Starfleet, and later disappeared.

Some viewers have noted that this does not seem to fit in the official Star Trek timeline: if Edison received command of a ship after the Romulan Wars, how could it be a warp 4 ship? By the time the wars had ended, the Enterprise NX-01 and the Columbia NX-02 had already been flying with warp 5 engines for many years. So this is a potential error on the part of the writers of Star Trek Beyond.

However, there is a fitting explanation for this. According to the Star Trek timeline, warp 2.5 was achieved in 2142, and warp 3 in 2145. Since we know that the Enterprise NX-01 launched with an experimental warp 5 engine in 2151, the prototype warp-4 vessel U.S.S. Franklin must have been built somewhere in the late 2140s. This is sustained by several other observations: the ship is very close in design and size to the Enterprise NX-02 (relatively flat with the nacelles almost directly attached to the saucer section); it has transporters that were only used for cargo (Enterprise NX-01 was the first with a transporter that could also transport humans); and it still has conventional torpedoes for weapons (Enterprise NX-01 started out with these, but soon switched to photon torpedoes).

So the U.S.S. Franklin must have been used as a cargo ship long before Edison received command of it. By that time, it must have been at least 11 years old, capable of only warp 4 and without transporters for personal transfer. Since Edison was already upset about losing his job as a MACO, being offered an obsolete ship may have added to his frustration.