Pike calls her a cadet when he asks her if she speaks Romulan. However, Spock later calls her "Lieutenant" meaning that upon graduation, that's her rank (Kirk is also a lieutenant--when the transporter chief is trying to beam him and Sulu up from the drill platform, the display reads "Lt Kirk"). It's not uncommon for academy grads to have a Lieutenant (j.g.) rank upon graduation, from what i remember from other series.
has a crew of 1000ish, most of whom are cadets (because the ship launched early, I'm assuming most of her assigned officers weren't able to make it back to Earth fast enough to take their postings), chances are it's almost entirely a cadet crew--and unless we find out somehow those 7 fleet ships had time to get shuttles/pods away (doubtful, given they were destroyed the second they came out of warp, until the Kelvin
years before which actually had time to abandon ship) the only
survivors of the academy graduating class of 2258 are those who survived aboard Enterprise