Regarding the role of Houses in Klingon politics, "Apocalypse Rising" makes it sound as if backing off from an ongoing war won't happen on Gowron's word alone - the Houses will have to be persuaded to follow their formal leader in a lengthy process. This may just be Gowron blatantly lying to keep the war going, of course.
But speaking of "Apocalypse Rising"...
I can't imagine why the Klingons wouldn't declare war
I'm not sure why Klingons would even believe
in declaring of war. It's not as if they would believe in peace in the first place - so why celebrate an imaginary transition from a nonexistent state to the normal state of affairs?
why did they declare war on the Federation?
They didn't. They just took "back" ownership of Archanis and said that if there's any attempt to resist, "there will be war". No actual declaration thereof was heard. By the next episode, the fighting is on...
Klingons and Romulans may or may not have had an alliance at one point - but they have definitely been fighting basically all the time, alliance or no alliance. All the way back in 2271, the two clashed at Klach D'Kel Bract, if Kor's story from "Blood Oath" can be trusted. Apparently, a little fighting every now and then keeps alliances fresh, in both Klingon and Romulan opinion.