Also a cruiser:
For "everyone?" The science and medical officers sure as hell don't. That's a full third of Starfleet's personnel that leave out that specialization entirely.
In fact, the color coding of their uniforms should actually be a pretty big clue as to what's really going on here. It's entirely possible that advanced combat is only available as part of the Operations Track, while command-level officers only get basic.
Yes it is. Starfleet introduced the concept of "court martials" as part of its review process back in the 22nd century when it was UNQUESTIONABLY a non-military agency. From Kir'Shara
REED: Have you told Starfleet you're planning to warn the Andorians?
TUCKER: No.
REED: Have you any idea what the Admiral will do when he finds out?
TUCKER: I'll save you a seat at my court-martial. Is there anything else?
And later in "Affliction"
REED: Captain, There are some obligations that go beyond my loyalty to you and this crew.
(Archer gestures the MACO to leave them alone.)
ARCHER: What the hell does that mean?
REED: I can't say any more.
ARCHER: You haven't said much of anything. You've told me a lot about your father, his years in the Royal Navy. Their tradition of honour and service. How do you think he'll react when he learns you're facing court-martial?
REED: I wouldn't know, sir.
This is Earth Starfleet, again. They make NO secret of not being a military organization and I'm not even sure they would classify as paramilitary at this point. Especially in light of their Season 1 depiction:
REED: I've been thinking about a ship wide emergency alert. Something a bit more comprehensive than battle stations. We're taking far too long to react to potential threats. With a single order from you, or an impact to the hull, the plating could be polarised, weapons brought online, critical systems secured.
ARCHER: I appreciate your concern, Malcolm, but this isn't a warship.
REED: That's obvious, sir. During our last run-in with the Suliban we were unprepared for their boarding parties. When the Mazarites attacked they disabled our aft sensors with their first shot. The list goes on.
(He hands over a PADD.)
ARCHER: I can see that. Run this by the senior officers, get some feedback, and we'll talk again.
REED: Yes, sir.
ARCHER: And Malcolm, don't call it battle stations. Think of something less aggressive.
I know this is just crap worldbuilding, but seriously... don't tell me that this is a military organization whose most advanced ship doesn't even have a written procedure for "What do we do if somebody starts shooting at us?" They're professional scientists (sort of) but amateur soldiers at best.