I have not played the Stormcloak side of the civil war, so I have never conquered Solitude with them.. but if you do, does the Emperor's cousin still marry in Solitude, and does the Emperor still go there on a visit/inspection at the end of the DB questline? It would be strange if he did, considering that the city is controlled by rebels.

I would think both events would still have to happen in order for you to play through the DB questline properly, should you choose to do that. I've only done those quests on the one playthrough where I joined the Legion so I don't know if there's alternate dialogue attempting to make it seem less weird if you've already taken Solitude for the Stormcloaks.

Yes but there is some changed dialogue between some penitus guards where they say they can't believe they let the Legion reach Solitude which kind of implies the marriage is possible some form of attempt to create diplomatic relations with the new country.

Civil war is so buggy, best ending for the game and story would be to take the seaon unending quest.

Well, there's still gonna be Imperial supporters in Skyrim even if they lose the war. But to answer your question, yes, the Dark Brotherhood quest still goes off without a hitch even if you finish the Civil War with the Stormcloaks.