Marisa,
I don't know of how much help I can be. I don't have kids, and generally I find men to be completely clueless, not to mention useless.
But, I do know that fighting, mood swings and tantrums totally pollute your environment and makes everyone who lives there feel [censored] and horrible. One of things that I have discovered when it comes to fighting with my partner, is that if we are constantly fighting about the same thing, it means that we aren't in a good place to talk about it. ( I know that, that sounds like a moron-easy concept, but it really isn't.) If it were, we wouldn't continue to try talking about it and failing miserably.
I have two suggestions:
1) If you are continually fighting about something, agree that you are communicating badly together, and stop talking about that issue until you are in a better place as a couple. This really doesn't take much. (I'm not talking about a few months, I'm talking a few days to a week.) And during that time, your ONLY agenda with him, should be to spend some nice together, doing quality things that you enjoy together, creating a sense of emotional intimacy. Once you have done that, you can try talking about it again, chances are because you both feel good about being together, you will be kinder to each other and more receptive to one another. The dialogue probably won't be perfect, but it will be a definite improvement.
2)Something else to try: both of you sit down and write down (one or two ONLY) things about each other that Needs to change. But on the same piece of paper also write down 4 or 5 things that the other person does for you, that you adore. That way, when you exchange lists, you always come away from it feeling loved and appreciated, not *****ed at or hurt.
~Hope this helps~