Will using the same headphones between my laptop and MP3 player cause them to short out? The headphones in question have begun to do that thing where they don't produce sound unless you bend and twist the cord a certain way. Could my devices have caused it, or are my headphones just old?

Its not your devices. This is because the headphones are not good quality, and what has happened is that a break in one of the many copper wires inside the protective coating has broken. So the effect of twisting or bending the cord makes the wire come into contact again and then not. This is not a solution unless you can make the wire NEVER move once you get it that way. Your best bet is to either buy another set, or buy a more expensive set that will last longer.

The other option you have is to determine where the break is, and repair it yourself. If you don't have a clue as to how to do that, then your better off buying a new set.