I'm thinking about starting the Imperial questline of the Civil War but the reasing I'm doubting is the fear of immersionbreaking dialogue.

I already started it once and wass bothered by for example children not moving out of the Jarls houses or the Imperial Guards that walk along the road with a prisoner calling me citizen and saying it is non of my buisness. Now for some reason I still want to play it (I already plan to kill the two sons of Laila law-giver since they're the only children not moving out of their house) but I want to know how much the Imperial dominance, or the end of the civil war, will be recognized by other characters.

Will citizens of Windhelm and the other captured locations speak of the end of the war and the new Imperial law-givers? And are there still dialogue options for how the civil war is effecting their lives or something like that?

There is some new dialogue and changes, yes, but nothing drastic. People do comment on it though.

You'll run into some npcs who talk as if the war is still going on too unfortunately.

At least if you beat the CW for the Stormcloaks, the Thalmor stop patrolling Skyrim and the Penitus Occulatus talk about the Stormcloak victory during the wedding.

No, the Thalmor do not stop patrolling Skyrim, it is something that Bethesda wasn't thinking about.

There's a difference between Thalmor assassins and Thalmor patrols. I have beaten the CW for the Stormcloaks multiple times and have stopped seeing Thalmor patrols with prisoners after the Stormcloak victory