Leonard 'Bones' McCoy: We've got no Captain and no First Officer to replace him.
Kirk: Yeah, we do. [Sits in captain's chair.].
At the end of the film when Kirk is being awarded his medal, behind the Federation Counsel you can see 3 flags in the background- the center flag is the California state flag. After Kirk shakes Pike's hand, and the camera pans up and back, the California flag is hanging significantly different so that it is no longer recognizable, seeming almost blank white. As the ceremony is indoors and all attending are standing motionless, this flag should not have moved.
Mr. Scott has a pet tribble, it can bee seen in his lab on Delta Vega in a cage by his desk. If you listen you can hear the signature sounds of a cooing tribble.