Well, we do have one canon example of a specialized ship for ortillery: The strike cruiser in Sup9.IMHO, destroyers and escorts are, courriers aren't, scouts and minesweepers probably not (in fact, the scouts I envision are very lightly armed, if at all, to avoid some maverick capitain to engage combat, even with the lightest forces). About ortillery, as stated above, I see it as a mission of the main fleet squadrons (and its fighter complement), not as specialized ships.
By the way, since you also mention them: What do you see minesweepers being or doing? It has never even occurred to me that there would be such vessels because I thought that even if there are space mines of some kind, regular escort ships would fight them.
The types of auxiliary ships I'd have are:
- Replenishment vessels (wet, dry and dromedary)
- Transports (troops and materiel)
- Repair ships
- Hospital ships
Other support ships:
- Couriers (often handled by multipurpose escorts)
- Scouts/Pickets (ditto)
- Bombardment vessels
- Armed merchantmen
Regarding hospital ships, you'd also want to have them because of wartime humanitarian concerns. Few of the Imperiums prospective enemies are so bitter or so callous that they'd ignore a suggestion to mutually agree not to target hospital ships. They'd be separate from other auxiliaries, and they'd be unarmed for the same reason.