Actually, he explicitedly used NASA logic, which he's not entirely wrong about -
No, his exact words when interviewed about designing the Enterprise were 'airplane logic." Thats why I put it in quotes. Matt Jefferies was an aviation artist, so he used what he knew about airplanes as he worked out what the ship would look like. NASA had nothing to do with it. In fact, GR's instructions to him were "no fins and no rockets" which would kind of leave NASA (full of fins and rockets) out of the loop.
Thoroughly listed, but since the OP is talking about a patrol starship in fiction and not trying to man an AC-130 in real life, he's free to replace as many of those enlisteds with junior officers as he likes.I can by that some college level credits would be required - I think a whole undergraduate degree is overkill for many positions (logistics, security, entry-level technicians that sort of thing).
On the other hand, his logic mostly works for the crew of a runabout, response vessel or even civilian freighter that spends most of its time docked at a station or at least has most of its maintence and support provided there. However, even there with a crew of five - based on a cargo transport - you're looking at three officers (pilot, co-pilot, weapons) and two NCOs (engineer and loadmaster). Interestly, the attack variant of the C-130 also gives an alternative answer to the OP as it has a total crew of 13 (pilot, copilot, navigator, fire control officer [tactical], electronic warfare officer [ops]) and 8 enlisted (flight engineer, TV operator [science/operations officer], infrared detection set operator [science/operations officer], loadmaster, four aerial gunners [armory/general duties]).