I've soldered on cheap $3.00 headphones all the way up to expensive sennheiser studio cans. It can be done, you just need the right tip on a decent, not necessarily expensive soldering iron. A low end weller works fine.
Well, if you cut just the wire, you would cover the splice with the heat shrink. If you get a new plug, you should cover the wire to plug junction but you'd need 3-1 or 4-1 shrink rate for that. I just shortened a headphone type wire for my sirius install and it wasn't hard at all even with the...