Oh god OP - I'm sorry, but this man sounds utterly hideous.
You say that you've been with this man for a long time, so some of his behaviour might seem normal to you when it really, really isn't. I don't want this to sound patronising, but just wanted to highlight why you really need to break from this "man" and stay separated. It would be easy for you to fall back into the trap of his empty promises and lies. Once he knows that you have left, he will be desperate to have his "dog" back.
Let's break it down:
He believes in this "alpha" crap
References to "alpha" is a reference to power. He is saying that he believes he should have all the power in your relationship. A healthy relationship is that of EQUALS. There is no space for a "top dog". It's also a term used by cretins who follow the "PUA code" or pick up artists code. It's a hideous, misogynistic movement that see's women as conquests and toys - not as people.
He isn't interested in your opinions and doesn't want you to assert yourself
You openly admit that he "shouts you down". This is incredibly unhealthy, and another very obvious sign that he doesn't see you as an equal. No-one, NO-ONE would treat someone they love and respect like this.
He has cheated on you twice, once when you were in a very vulnerable postion
Again, a total and utter lack of respect for both you AND YOUR CHILDREN. I'm sorry, but a "good father" does NOT behave like this. They are there to support the emotional and physical health of the mother. Presumably after the first affair you got the whole "I love you, please don't leave, it was a mistake" bullshit. Yet he did it again. When you needed him. He is utterly repulsive.
He see's women as objects
The stuff you found on his phone is at best FUCKING CREEPY, and at worst, potentially illegal. I'm pretty sure his employers would sack him for misconduct if they found evidence of him wanking into someone's shoes at work. And what about the poor owner of the shoes? Him seeing a woman in that light is abnormal. It's objectification at it's height.
He demands sex
So despite him treating you like shit, he still expects you to "service" him. Because to him, that's all you are there for. To look after his house and home, and provide one outlet (but not the only one...) for his sexual urges. He doesn't care about your feelings on it - very dismissive of your reasoning, because to him, your feelings and opinions don't mean jack shit.
He will never admit to being wrong - UNLESS it suits him
You've demonstrated that he will blame you for everything he can get away with. When you actually leave, he will probably turn into a grovelling piece of shit that will promise you the stars. Thing is, he has already proved that his promises and words are utterly hollow. You would be a fool to believe them now after so much evidence to prove that he is a liar.
I know this must seem like the world is ending, but it isn't. Contrary to how you might be feeling, you don't NEED this man. You honestly don't. If anything, you need to be away from him and to re-build your own life. There is light at the end of the tunnel, but you have to be prepared to actually walk through the darkness to get to it.