Chekov has a grey collar so presumably is science officer. Saavik in TSFS wore a white collar but I think that reflected her role as a mission specialist rather than a part of the crew as first officer.Chekov is well-known enough for Marcus to respond to him on the comm, so there's nothing inconsistent with him being the XO. But if he was originally intended to be a Genesis project attaché, that would make sense too.
As for the rest of the crew, the impression the film gives is that they are attached to the ship under Captain Spock and are training the cadets. This certainly seems to be the case for Scotty.
Sulu's a bit different because his line "any chance to go aboard the Enterprise" suggests he's not a permanent fixture, and there more as a favour to Kirk. That never made much sense to me - why go on a training cruise but not actually let the cadets fly the ship? You could cut him from the film completely and it wouldn't have mattered at all. Same for Uhura.
Sulu should have been supervising and only taken over the helm during the crisis. The Trek movies tended not to serve the supporting cast well in any capacity. Even in STIII, they could have made something more of Uhura's role in delaying the security response to the theft. Sulu's upcoming promotion to Excelsior could have added an important layer of pathos to Scotty's sabotage.