This is a venture into the unknown. What would investors think? That there must surely be a solid gold planetoid on the other side, and thus just send in a mining ship? It's a big unique phenomenon that no one knows the nature of, and thus probably contains information of unknown utility. Since anything could be inside, having all the science bases covered seems wise. If they wanted to survey a new mining claim, they have plenty of regular, non-event-horizon-covered space to explore. The spirit behind it would have to be more akin to "let's put a man on the moon" than "let's go survey the Congo."
Governments probably are structured like coalitions of more independence given to individual planets due to scale. It's that whole "months to cross the Atlantic" thing. Let's say FTL here (the one convention, since it's needed for this kind of coalition in the first place) requires mass to work, hence the inability to simply send signals faster than light.
Engineering is probably the same as manufacturing because that's where all the tools for the engineers to make stuff would be anyway.
Having dedicated biology labs, archeology labs, materials labs and other stuff smacks of knowing exactly what we'll find on the other end, which is another region of space with life and available resources, which, as far as I'm aware, the investors would not. The mining and fabrication equipment is for adapting the ship to make a return trip if the return trip requires additional propulsion and ship repairs, obviously, and that's if they happen to go to a region of space (which could not be the case) with resources to be had (which is even more in question).
Way I see it, point of the mission is to discover whether the Dark Star is actually a wormhole (which I assume is how anyone convinced an investor to back it) and whether it goes two ways. If either is not the case (like the ship merely being elongated and shredded beyond the event horizon or there being no way back), then further science performed becomes a moot point due to inability to relay information back. It's a proof of concept, sort of like how putting a man on the moon is a proof of concept. When they put a man on the moon, they planted a flag, grabbed a whole lot of rocks and cheesed it back to Earth. They did not start building a moon colony, since that would overly complicate the mission and there was no pressing need to do so.
Most of the biolab, archeology lab, two types of each lab, let's have space guns business runs under the assumption that the Dark Star is definitely going to be survivable and that it's going to lead us to uncharted intragalactic space teeming with space age life with roughly equivalent technology to ours, and that it goes both ways, too, despite that being a string of increasingly unconvincing assertions from a semi-realistic standpoint. We know from metagaming that all the space opera things we like are definitely going to at least appear on the other side, because otherwise it'd be quite a weird space game (though I can't say I'd mind it taking after something like Dark Star the film instead), but in-universe this is just an unproven concept that is astronomically unlikely to be workable even before you put expectations of life in there.
That's if the mission is funded by governments and reasonable investors, of course. If it's funded by, say, an insidious cabal of crazed galactic trillionaires or a stupendously wealthy interplanetary cult, I could see it being built as a colonization ship rather than an exploratory vessel, since they might not be expecting a return at all and could simply have done it as a vanity or dogmatic project without a trace of professionalism or reasonable expectations. The cult, too, would explain the volunteers. Point is, underlying motivations are important.
Also, I suspect it'll be more interesting if we have none of those extraneous laboratories or combat capabilities when it turns out the Dark Star has stranded us in the distant past/future of the universe or in a considerably scaled-down alternate universe or uncharted space or outside the universe or something of that nature.
Making it so that the ship doesn't function without the AI is what I would describe as bloody stupid. Secondary backup systems utilizing non-sentient computers are quite possible. If that isn't possible for some arbitrary reason, DON'T BE THE FUCKING AI. Seriously. Every third post here is you saying "You can't do that because I'm the AI"[<-Paraphrased]. Granted, half the time its because boarding hooks in space or some other asinine idea, but its still getting pretty fucking old.
Well, that's easily fixable. As mentioned, make the AI into oversight for less advanced systems. Really, the reason I mentioned the AI as running all the ship systems is because I'd like a fewer number of moving failure-capable parts. This could even have been the plan all along, since if we imagine the AI as an organic analogue, the non-sentient computer systems running background processes would be vegetative functions without any input from higher processing centers. Or just have Kevak's AI be the redundant system there in case of the ship's automated systems failing, as mentioned. It doesn't really matter all that much.