A:AnswerNo the cord cannot be replaced. My cat chewed through my cord and I had to take the headset apart an sauder a TRRS jack into the hole where the cable was. If the headset is damaged I doubt you will get anything for it.
A:AnswerIf your 360 controller has a 3.5 mm headphone jack, it should function as a headset, but don't expect to use the mic or change the volume. It's designed for the Xbox One and has a specific connector module that allows for all the extra gubbins to work.
A:AnswerI don't think you'll have to. Everything you need should be included in the box and the set up usually only requires you plug the headset into the controller and update the controller how you normally would by connecting it to the xbox via that little chord.
A:AnswerThey run from your controller so as long as your controller is charged it works. I can go 2 days without charging my controller and I game alot.