Spock is off on his own to rescue the 2 without saying anything to Kirk. We learn that he hasn't really dealt with his grief and has been acting somewhat recklessly since Vulcan was destroyed.
That sounds to me like it's setting up a character thread for the movie. Particularly considering the mission Spock is on at the start of the movie.
Indeed. To quote Uhura from Countdown 3, "Spock risked himself unnecessarily. He did it before, when the Galileo crashed. It's like he's looking for any opportunity to sacrifice himself."
That seems to fit well with the Volcano incident.