At work I fought with my boss...?

To answer your First question it is not going to help by thinking about it. Things happen all the time based on the facts you provided you seem to be right. Well I believe every relationship builds with respect,love and time. First thing you do is from now on is do not yell. Always smile at him from now on. You are not going to lose your ego if you can let him know that day you were not in a good mood and hope it did not hurt. Just mention only that from your heart. Time will build everything slowly. This is all done because you are a human don't worry if he is your boss or your junior. Just because he yells at everybody it doesn't mean you yell at him back. I Feel this is a good way to handle this issue. It is left to you to decide if you want to do it.