A "classic example" is a stereotypical example. There can be irony loaded into this phrase depending on tone of voice. It can also just be a straight assertion that this is a common and very representarive example.

They are close to the same meaning. To say "I will go tp new york next year for sure" means you are very hopeful that you will go to new york next year but implies maybe a little doubt about it that you are trying to persuade yourself or your conversation partner that you might not go. I'm sure I will go to New York next year sounds more certain.

Yes