Hello Hopeful....I'm so sorry that there is so much drama going on this evening. Try to breathe, and slow down, and realize that Rome was not built in a day.
Perhaps after everyone has had a good night's rest, then you can all sit down and draw up in writing a firm set of boundaries. This can include where you will take him if he is found with alcohol again--homeless shelter, Salvation Army, hospital, wherever you mutually decide. You might decide that he does indeed have to leave tomorrow, but none of this has to be decided right now--tonight.
Perhaps another plan can be discussed tomorrow about what he intends to do to support his child. If it were me, I would make it very clear that I would not provide financial support or free daycare because he and his girlfriend are perfectly capable of doing so themselves.
I'm so sorry for your evening, and I hope that rest comes for you all tonight! Please do take care!!