1. If someone gives you a negative trust, can you fight back like proving yourself correct and removing the negative trust from your profile.
Reporting you doesn't mean they can actually give you redtrust as long as you haven't done anything wrong nor did anything that disobeys the forum's rules and regulation. And even reporting someone whom you'd knew reported you would even make a single change with his/her reports. DT and management often checks first the validity of the report and therefore no one other than them are capable of giving you redtrusts.
2. what advantage you get if you give someone a negative trust?
Ofcourse the huge disadvantage is that having a redtrust is the disadvantage itself. LOL. Trust is a huge thing in the online world, and having a mistrust simply means you cannot be trusted thats all.