I figured it was a slipspace portal, like the ones the Guardians come through, the one from Halo 3, and the many on Requiem in Halo 4.

: What is the purpose of genesis? Just an interface to the domain?

I think Exuberant Witness touches on it, but it's not detailed unless it's in the Forerunner terminals. It's some kind of refuge, and I figure it must also be a sort of home base for the Guardians.

: Why does exuberant help osiris? Why is she against the application of the mantle when she was presumably created by forerunners who used the mantle?

Well she argues with the Warden Eternal, saying the "Created" are not the rightful Reclaimers, but she does also describe the Mantle in negative terms.
From what I understand the Forerunners basically divided their society by discipline or industry, and as with humans people who work in the arts are different from people who work in the sciences who are different from people who work in engineering or hospitality or academia and so on. Genesis was associated with Builder rate of Forerunners (per a line of dialogue from Exuberant), while the Mantle would presumably be more the thing of the ruling class, and maybe the Builders didn't care for it. The Librarian, who was a Life-Shaper and so maybe had a hand in it as a refuge world, also didn't seem too enthused about the Mantle.
From the little audio terminals it seems there are living Forerunners there too, or maybe Forerunner AIs. Maybe they decided they had been wrong.

: Is it supposed to seem like cortana has been somehow corrupted/changed by the domain? Because she really doesn't seem like the same character at all any more.

That is the question, isn't it? I think we won't know until Halo 6. I agree though that it's not just different, it's out of character, and there's not the direct explanation this time that she's just not in her right mind, she says her rampancy was cured and she acts stable. It's the weirdest and worst part of it to me.

: This is exuberant's question, but I don't think it was answered: why did cortana recall all(?) the guardians to genesis? And what was up with the implication that some of them had returned automatically? But she literally just sends them all out back to populated systems afterwards. Why call them all together at all?

I'm going with Morhek on this, resource management. You'd want to know how many there are and have them all activated and equipped, and then send them where you actually want them, which might not be where they were originally. As for automatic activation, your guess is as good as mine. I thought it was going to tie back to the Didact (the new old one), but he's completely forgotten in this.

: Why is blue team so much less comfy than noble team?

Noble team might have had thin characterization, but what we get for Blue team in the actual game is even thinner (that and Jorge was a big teddy bear).

: Why did they add ": Guardians" to the title?

I think it's because people tend to look down on things with just a name and a large number in the title. I think they're going to mess with the numbering in some way or just remove it.

: Where's my comfy poncho chief?

It is a mystery.

Here's one possibility:

Josh Holmes: "Guys, I'm not feeling so good about this. We've got shoulder charges, ground pounds, hold left trigger to zoom, infinite sprint, the name 'guardians', a cloak over top of armor...."

Frankie: "I know, we were trying to get away from Bungie--"

Sparth: "We even fucked up Jackals and caked things in mud to do it."

Frankie: "--but we keep coming back to their stuff. I mean, we're even adding color and humor and understandable enemies back into the game."

Josh: "Hmm...."

Frankie: [ponders in Scottish]

Sparth: "This might sound dumb, but let's get rid of the cloak."

Josh: "No, it's brilliant."

Frankie: "Yeah, it doesn't really go with the sty--"

Josh: "The ground pound is way too essential."

Frankie: "Oh... that too."