Sorry for my late answer.

Yes, we have "adjø," presumably from "adieu."
Bokmål and nynorsk have the same status, but bokmål is preferred by more than 90% of the population. I've never used nynorsk in any real-life situations. We all have to learn both in school, which is largely a waste in my opinion. Bokmål is in a way "less Norwegian" because it's very influenced by Danish. Nynorsk was devised by Ivar Aasen in the 19th century. He literally travelled around the country for years, gathering information about dialects (he avoided large cities because that was where Danish influence was largest,) then he put together a new writing language, using words he had picked from various dialects. Learning to read nynorsk is one thing, but writing it...! Nynorsk was somewhat hard, because it's often easy to choose the wrong word (i.e. a bokmål word instead) because they're so similar.