You start by acknowledging everything you did wrong. You gave her silent treatment, that's a big no-no. That's emotionally withholding from your girlfriend and is abuse. Another thing is if you want this to work and progress, you are going to have to open up to her. I know it's not easy and everybody opens up at their own pace but if you do not talk about what's bothering you, she will automatically take it personally. If she doesn't, more power to her for her self confidence and comfort in her own skin. When we care about somebody and they treat us like that, we think it's something we did wrong. You would feel the same way if she did it to you. You are also getting angry and hitting things and being rude to people. I don't know what your girlfriend really sees in you, but this kind of behavior shouldn't be tolerated.
This kind of behavior is actually pushing her away and she will eventually want to be with somebody else if it continues. She may feel attached to you right now but I'm telling you that if you don't make a serious effort to change your behavior and curb your attitude, you will lose her. I'm actually shocked and amazed you came on here looking for help while you still have her. You found this cancer early and you have the power to make your relationship stronger and better.
You sound like you are worried about how you appear to others. You want to know if you are being a good boyfriend or does it mean you care? How do you FEEL about it? What does being a good boyfriend really mean to you? You may at times show you care but at other times you act in total disregard to her. Even if you meet the bare minimum requirements of being a boyfriend (not cheating, not physically abusive) you will still lose her. I promise you.
You guys are very close to losing it and even now I can't guarentee that you will be able to salvage it. It has to start now. You have to be self aware of your own behavior, you have to understand how to handle your anger and jealousy and find ways to vent that on your own without mistreating her (ignoring her, yelling at her) in the process. It takes some serious growing up and the sooner you do the better off you will be.
Waking up next to a beautiful girl,
Step outside and say hello to my beautiful world.