Upon meeting Haskill for the first time, he comments on the nature of the Shivering Isles door in saying that it does not represent a threat to Tamriel, and thus "no compact has been violated".

I assume he is referring to the Compact made by Sotha Sil with the Daedra in 2920...but since the Tribunal is dead (or at least, 2/3rds of it) and the Heart of Lorkhan is destroyed, why does the Compact still stand? Is my assumption wrong and is he referring to some other compact? And are there any hints from the Sermons that relate to what Sotha Sil promised?

I think it's more likely that he refers to the "barrier" to Oblivion itself - though the 2920 contract might also play a role. In essence, it's the explanation for why a gate like this is open even after the Oblivion Crisis was averted: it's an invitation, not an invasion.
Actually, we don't know if the 2920 compact still stands - that would depend on the exact terms, which we don't know (aside from what's mentioned in 2920, e.g. summoning rules). Actually, that's one of the things that would like to know.

Initially I thought that it referred to the Alessia-Akatosh pact, the one with the Dragonfires and all. Doesn't entirely fit in hindsight, though.

Well, I think a consensus would be that Haskill refers to the various rules which govern daedric access to Nirn. In fact, human access to Oblivion wasn't really limited at any time - it was just very difficult and extremely dangerous, as can be seen by the example of Morian Zenas. Some of the easiest accesses actually were the shrine quest of Boethiah (also an invitation) and the access to Magas Volar via the Daedric Sanctuary Amulet in Tel Fyr (invitation or at least the voluntary act of a mortal wearing it - I wonder whether that's mentioned in its user manual, though).

I thought that, too, until I realized (and this really could just be game-mechanics) that he says the same thing regardless of your position in the Main Quest. Whether the Dragonfires are unlit, or whether the Dragon Statue stands, doesn't matter. For a period there, there is no barrier to Oblivion (hence the invasion), but he still mentions the compact. I suppose that could, as I mentioned, easily be written off as game mechanics, but I think it is still at least worth someone who has looked over the Sermons and other Tribunal-texts a lot to consider if any of them contain any oblique hints as to the deal Sotha Sil made.

Mehrunes Dagon may have broken the contract, but Sheogorath still didn't. He didn't need or want to either, his portal was pure invitation for a mortal champion to come to his realm to help him with a problem. Mehrunes broke it, because he has to, and the other princes are opportunists.

Might be good to separate all the different rules involved.

0. Anything can travel unless otherwise limited. This is the basic assumption to keeps everything simple.

1. The barrier keeps the Daedra out. This is generally speaking, anything from Oblivion with a soul. The barrier does not hinder Mortals leaving or entering. Both can be derived from Liminal Bridges, Doors of Oblivion and the Clan Mother.

2. The barrier can be overcome or side stepped altogether. Think the Princes Birthdays, the Oblivion crisis, summonings, large acts in the alligned sphere as in the Aldudagga, ect.

3. The Daedric Princes can only be summoned by skilled conjurers. This is the gist of Sotha Sils pact in 2920. I reckon it is still in effect. Though Sotha Sil thought otherwise, Mehrunes had been summed by the Witches.

So in general the rule is that for Daedra to enter, some event must have taken place. For anything else, it doesn't matter. It's a really practical point as it keeps the Daedra out unless the plot requires them and constrains their power so they can't easily destroy the setting.
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Considering that the Portal to Sheogoraths realm was not a Daedra in any way or shape, but only provided a way for a mortals to leave Mundus and enter another realm no rules where violated.

Summoning opens a temporary portal sufficient for a single Daedric item, creature or person to cross to Mundus. It appears to be the inverse of the doorway to the Shivering Isles and similar in its conformance to the rules, in that it's an opening into the realm of the being who cast it, and not into anyone else's. I.e. you can invite an outsider in your realm, or even summon one in, but you may not use a portal to invade another's.

What's a grey area for me is whether a visitor to SI is breaking any compact if they allow or invite a companion who is of that realm (e.g. Mazken or Aureal) to join them in returning to Tamriel. The summon spells you get from the SI main quest don't allow it, but you can pick up followers who can trsit the doorway. The portal used is not theirs, but the invitation to another realm is coming from a denizen of the entered realm, so does that make the transit OK? If that's "legal", then do the activities of the Mythic Dawn legitimize the opening of the Gates to Oblivion? Did Martin break the rules by opening a gate into Markur Cameron's Paradise?

Invitation is a nice way to phrase the effects of the portals and summoning. However Mortals, rude little bastards, don't need an invitation. Doors to Oblivion mentions that you can enter Oblivion by merely starting to be there. So Martin did not do anything he shouldn't have been able too.

In fact, Daedra WANT us in their realms usually. They get bored with the predictability of the univers outside Mundas (except for Jygglyag) and like us to be there. Those that don't simply don't set rules on their realm that mimick the earthbones and let us live, or don't allow us to enter. Mehrunes wanted us to enter Mankar Cameron's realm, and it was made with his power.

I think Mankar Cameron's realm is different from the deadlands, and serves as a reward to Mankar Cameron for being the Mortal hand of Dagon on Nirn, where he cannot normally go. And I get the impression that when he DOES go there, he MUST do it by breaking something. I think the Gateworlds are something special, and several of them have fortifications and otherwise great defensive locations, even from a siege standpoint. I think Dagon expected resistance, just not on the scale of one or two squads, if that. The fire towers and mines would be hell for a slow moving siege force. I am under the impression that the reason so few daedra are actually stationed there si either a) game mechanics or B) arrogance on the part of the local commanders. (At least for those worlds that don't spawn as strike points against cities).

Indeed they are. Keep killing my puppies.