Translation:He will soon realize that he is making a mistake.
Jeg har rett og du tar feil.
(Pretty similar to German: Ich habe recht, und du gehst fehl / liegst falsch.)
Not in this meaning, no. Not that you won't hear it, but it's still incorrect.
You can use it in other contexts, but then the translation would be "you have the wrong" in English.
It can mean both "want" and "will", but in this context only "will" makes sense.
Would "Han skal snart innse" be grammatically correct, and if so, would it seem unnatural to a native speaker?
It's grammatically correct, but would imply that the speaker/writer is planning to have a hand in bringing this realization about. "Skal" implies intent.
I suppose so. It also gives 'shortly' as a meaning for snart. When i think of the differences between shortly and quickly, I think of quickly as movement or quick thinking (which would imply intent?) but shortly is just temporal.