One thing that I don't get is this. Why do the AQ Allies feel the need to piss of the Dominion.
The Dominion told them not to go the Gamma Quadrant. Yet people like Sisko feel the need and entitlement to go there.
It was actually Jadzia who had the "we can explore where we want!" line of dialogue. On the one hand, it's inexplicable that the Feds feel the need to explore on the other side of the wormhole when, if the wormhole did not exist, they would simply continue their regular pattern of exploration and be none the poorer for it.
Was outer space in the GQ so much nicer than outer space in the AQ? Were the planets different?
Star Trek has never implied that different sectors of the galaxy are noticeably different in the proportion of habitable planets, which is always at an unrealistically common level. (If habitable planets were depicted as being rare and valuable, then all these territorial squabbles would make a lot more sense.)
Another big open question: did the GQ end of the wormhole terminate inside claimed Dominion space? If it's inside, end of discussion. The Feds cannot use the wormhole and the Dominion has every right to force them to stop. The series never once clarified this crucial point.
On the other hand, the Feds have often, perhaps inadvertently, been depicted as idiots who go barging around in other people's claimed territory and therefore cause the fights that they keep getting into. So the way
DS9 showed things isn't all that inexplicable. And the final peace-treaty scene did sorta acknolwedge the Feds' partial responsibility for the war (which is the first time the series even broached the subject, better late than never I suppose).
but the fact that it took two years before Dominion first contact shows that Idran is quite some distance away from Dominion space.
All that proves is that the Dominion claimed a large amount of territory. The size required them to patrol sporadically which meant by the time they came across interlopers, a violent response would be required to expel them from their foothold.
Not really nice, I'll admit. It would be nicer if they claimed less territory and gave more frequent and more kindley warnings to interlopers. But space is a big, nasty place and it's not difficult to predict that territorial disputes will be resolved violently in many cases. The burden is on the colonists to put more effort into ensuring that they're not settling someplace that will get them killed. Neighboring worlds certainly would be aware of the size of claimed Dominion space. The Bajorans just didn't check the place out thoroughly.
In any case, the Feds could have taken the hint: the Dominion does not play nice. Isn't it better to stay away from the folks that don't play nice? Is the AQ all out of explorable space? Starfleet acted like they were looking for trouble - possibly putting on a big show of force to impress the Bajorans, who clearly could not defend themselves. Demonstrations like that is how the Federation expands - by violent confrontation in defense of members and would-be members. Not too different from the way the Dominion protects its territory by violence.
A quadrant of the galaxy is a big place, and the GQ doesn't belong to the dominion, not in entirety even if much of it does.
Why can't a single power claim an entire quadrant? A single power can claim a whole galaxy if they have the muscle to back up that claim. Is there some rule book for the Milky Way Galaxy I'm not aware of?
And there was no indication that the Dominion claimed the whole GQ - just the space around the terminus of the wormhole.
However, that said, the people who are saying that the Federation provoked the Dominion by going through the wormhole probably need to watch the show again, because as other people have rightly pointed out you've probably missed the bits where New Bajor is wiped out for no reason other than it's there,
The colony being there
was the provocation. All the other violent acts were precipitated by that provocation. The Feds should have been smart and said, "shit these guys will go berzerk just because of a colony, we better declare the wormhole off limits before this gets any worse." Imagine how many millions of lives they could have saved. What exactly is on the other side of the wormhole that is worth that?
And note the Romulans were the smart ones, early on. When they realized the dumbshit Feds would continue to provoke the Dominion, they did they only thing they could do to defend themselves: blow up the wormhole. Only after that failed did they move to Plan B with the Cardies, and that ended up making things even worse.
The Feds, Rommies and Cardies all played a role their mutual fuck-up in handling the Dominion situation. To the Doms, all three were the same - just AQ clowns who needed to be smacked down. Even if the Feds thought they had a god-given right to the whole galaxy, they should have been savvy enough to understand they were endangering other AQ powers, who might be therefore motivated to try dangerous and possibly counter-productive things out of frustration and desperation.
On the other hand, Dax's previous host was a former ambassador, and it would know laws about free space and exploration...I think...
What laws?

Fed laws, right? Why should the Dominion give a flying flip about someone else's laws when clearly they have their own, namely "stay the fuck out of our space"? The only law that prevails in outer space is the Law of the Jungle - whoever has the muscle to get their way is the one that wins and then parades their "laws" around like they should apply to everyone.
In the end, you had two violent empires clashing, regardless of what Fed propaganda would have you believe. They both had their ideology to rationalize their territorial conflict, and they both had good reasons to fight (largely for domestic purposes - both sides would prove to their subject worlds that they were big and bad enough to provide security, so they better stay obedient). In the end, they called it a draw. Pretty realistic and sophisticated depiction of warfare, I'd say.