I don't think it really matters. Quality headphones are designed for decades of use, I doubt even a few hundred hours of constant use would hurt them at all, at least as long as the volume isn't at max with an amp.
Personally, I don't think so, really, all music is just movements of the driver. I've heard people say that if you use the kind of music you listen to, it will "adapt" those qualities, but I don't believe that. What I would try to do, though, is use music that uses as much of the spectrum as possible, the super highs and super lows (some people just use sweeping noise). Doing that won't affect the qualities of the headphones after burn-in, but it is possible that it could speed up burn-in a *little* bit.
Volume does a little, because when you think of it, it is doing the same thing, but harder. Kind of like running in a new pair of shoes instead of walking to break them in. Though I wouldn't go too high, just maybe a little louder than you would normally listen to. Headphones are meant to be played somewhere around the (fairly vast) range of listenable volumes. They are not designed to be at max volume on your gigantic desktop amp.
I'm sure those with more experience and knowledge will expand on and critique my answers, so be sure to read as many opinions as you can, they'll all follow a similar curve, and the average is probably closest to the truth.
Also, don't let burn-in be a reason to not use your headphones. It's always good to use them and see how they change as they burn-in.
Enjoy your new headphones.
