No. The Nike+ only works with the iPod nano. You probably wouldn't want to run (at least strenuously) with a classic due to vigorous shaking of a hard drive is never really recommended.
If you're cycling, the Nike+ system won't do you much good anyway, as it communicates with a sensor in your shoe that monitors your steps, which likely wouldn't register on pedals (though there have been rumors that 24hour fitness might be adding equipment to their gyms that work with the Nike+ system.
I'll let someone else answer the headphones question, though if you do a forum search you'll likely find your answer.