It's definitely a drink problem inasmuch as his drinking is causing problems in his family yet he continues to drink to excess. He's also got a problem in that once he starts drinking, it seems that he cannot reliably control how much he drinks. Drinking to the point of puking is something that most people grow out of in their early 20s.
Whether it's a problem in a "is he an alcoholic?" kind of way is harder to say. I dunno. He may well have developed some kind of alcohol dependency but whether that's got to the stage where his only real option is to stop drinking entirely, forever, is impossible to call.
From his point of view, he's likely seeing it as him just doing what he's always done and that has likely become a big part of who he is. I could well believe he's thinking "Hell, Greatfun's put up with me doing this for 15 years, why's she got a problem now?" This isn't to say that you are not entitled to feel fed up with his atrocious and selfish behaviour; he's behaving like a twat. But it may not be easy for him to change.
More importantly, right now he doesn't want to. He's a grown-up who's not doing anything illegal. It's stupid, selfish, hurtful, disrespectful and immature, sure, but you don't have a right to insist that he changes his ways.
On the other hand, you do have the right to choose with whom you live your life.
One way of communicating that to him that may be worth trying is to avoid the "You did this..." and You did that..." but, instead, start with "I". Eg, "I am worried about your health", "I feel scared and angry when you come home drunk", "I'm sad that the children miss out on time with you when you're hungover".
And, maybe, "I can no longer see a future for us with the problems your drinking causes. If you don't take steps to deal with your drinking, I''ll be seeking a divorce." But don't say the last unless you're fully prepared to follow through on it.
If you're not ready for that last then that's ok. Either way, you may find Al-Anon useful. It's the "friends and family" offshoot of Alcoholics Anonymous. It's for people whose lives are being affected by someone else's drinking (they don't have to be an alcoholic). They're not going to try to persuade you to leave him and they're not going to tell you ways of getting him to stop. What they can help you with is to show you ways of getting your attention off his drinking (which, I know, can become almost an obsession) and putting it onto yourself and helping you find out what you want from your life.
You may also find a book called "Co-dependent No More" by Melody Beattie very helpful, as this is also aimed at people who are in relationships with people with drink/drug problems.