The dude sitting at Engineering station on bridge ? but he wore the same uniform as Saavik if I'm not mistaken and red collars stands for cadet rank.. not sure about that tho, if Saavik was LT she couldn't be cadet, could she ?
She was a Lieutenant when taking the Kobayashi Maru test. I think that's a sign of officer training in Starfleet. So yes, she was still a cadet in the fact that she hadn't graduated from the Academy yet.
To answer you baz, likely Spock did not have a first officer. After the Enterprise returned to dock after however many years Kirk kept it on the frontier, Starfleet finally MADE Kirk accept promotion, promoted Spock to captain, and then reassigned the Enterprise as a training vessel. Likely she was never intended to actually go on duty, so why would Spock need a first officer?