Will they only do it if the noise is present, plus you are still under factory warranty? What about if you have a non-Factory warranty on the car?

 

Seems to vary by dealership. Some want to reproduce the noise, some will accept a video, some will take your word for it.

And yes, to have it covered under warranty, you would still have to have a warranty in place (factory or non-factory). Otherwise, you'll have to pay for the fix (unless you can squeeze a goodwill repair out of the dealership - again, a big variable).