Please, please document this issue about him leaving a small child, a 5 year old, alone in the home. Write down what your child said, write down the fact that the babysitter corroborated the story and date it all.
Document incidents when he's been violent toward you, whether physically or verbally violent or both. Call the police every time he gets verbally violent or raises a hand to either of you. Tell your neighbors that if they hear shouting from your house to please call 911. Tell them of the situation, especially the one about him leaving your daughter alone, so that they can help you keep an eye out in case you HAVE to leave your daughter with him. Tell them to please call you immediately if it happens and then you can call 911 on him and also get home to your daughter.
Tell family and friends what's going on. The more people that are aware, the better so that if something happens to you or your daughter (heaven forbid) then the police know who to take into custody first. They'll have an automatic suspect.
If you're questioning whether he's an alcoholic, one doesn't have to be drunk to be drinking too much. Drinking too often and to numb the emotions are enough to "count." If you feel it's too much, then it's a problem. His rage and deflections are a sign of it.
If you file for divorce immediately, and file for full custody with supervised visitation, make certain that you include just how dangerous his behavior is. That's why documentation is so important. A court will not want your daughter to be in danger when she's there, or risk her life.
Your daughter WILL NOT hate you. Remember, she told you what happened because she didn't feel safe. She knew he was wrong. Even at five years old, she knew. And he's behaving as if he's been wronged. He's a danger. He's full of anger and he's not safe as a parent. It's not illegal to be an alcoholic, but his behavior is sure as hell illegal. It's also unpredictable.
Personally, in your situation, I wouldn't wait it out. I'd get a lawyer NOW. I'd file an emergency petition for custody, I forget at the moment what it's called but the lovely ladies here will certainly remember.