For your first question (the one on the title), modern "nigiri sushi" was only invented in the 19th century. Previous food also called sushi was rice-fermented fish, which wouldn't have been included due to kaiseki's emphasis on fresh, seasonal ingredients. For your other questions, it's basically "it depends on what that specific restaurant serves." Even if traditional kaiseki didn't include sushi, it doesn't mean modern kaiseki doesn't.

@Moshi It seems you are experienced with that stuff. Do you mind fleshing out a full answer to the post? I think a lot of users here (including me) aren't well versed in Asian cuisines (and their respective history) so the chance for a proper answer from regular participants is pretty slim.

@Zerotime I wouldn't say I'm well versed in Asian cooking either, but I'll try to write an answer