Lotus,
My personal opinion is that when you have reason to suspect infidelity (such as when he announces to you that he wants to be with other women) then that gives you just cause to check his phone and do whatever else you need to do to ascertain if he is in fact being unfaithful. Other people on this site may tell you differently, that trying to 'keep tabs' on an alcoholic is just going to drive you insane, but if he is in fact cheating on you then I think that is something that you need to know so that you can stop and consider what your choices are from there. If you have to look at his phone in order to know if he is cheating, then so be it. And frankly I wouldn't worry about what the other girls think - they ought to know better than to get mixed up with a married man, and if they do not, then having an enraged spouse to deal with is a just consequence for them.
I haven't heard anything in your post that implies that he wants to quit drinking for himself, only that he briefly stops drinking because you ask him to or he does not want to offend you. If that is the case then I suspect that he is not ready to quit drinking, period, in which case you can expect more of the same of this behavior. If that is the case, then I think transform said it best - you can walk away from this guy, or you can allow him to torment you with more of the same.
Flirting with other women, telling you that he wants to be intimate with other women, trying to demand sex in order that you fill 'his' needs, going out until all hours of the night, getting drunk, you name it.... this is a totally unacceptable way to be behaving in any marriage, and I hope you can step outside the situation enough to realize that.
One other thing just to keep in mind - between his emotional outbursts, his obsession over his comic book collection, going to fraternity parties, it seems like this man really is stuck in his teenage years. When an A starts using as a child or a teenager, his or her emotional development STOPS at that point. If your husband has been using that long, then this is what you are truly dealing with, is a child grown older, and I would take that into consideration when trying to assess if there is any remaining potential for this marriage to work.
Best wishes,
MZ