My suggestion would be to delete only the negative rep feature if possible. If peoples' reputation is going to be a feature that is available for public viewing, those that feel some pride about a big rep number are naturally going to be pissed when someone gives them ANY kind of negative rep. It's titled neg rep for god sake, and I don't think anyone here looks at em and says--"yup yup I deserved that one".
If no one was able to LOWER someone elses rep and pissing contests went on in PMs or something, I doubt that you would have as many complaints because the way it is now, rep is something that everyone can see and some folks want to protect thier reputation. A negative PM you can just delete and no one ELSE knows the difference.
Bottom line? Maybe the old adage that " If you don't have anything nice to say about a person, don't say anything at all ", is some pretty damn good advice and could decrease some bickering.