I am sorry this has happened to you and your family. It is always heartbreaking when a spouse strays in a marriage. It undermines trust and hurts everyone involved.
It sounds like your husband has put his alcohol use and his need for approval for it before you and your family. That is not uncommon with alcoholics but that does not make it less painful for you. The fact that his girlfriend drinks as well may be the thing that drew them together. But it is not the bases for any relationship, it is only a shared interest. With both of them using, they approve of each others behavior and do not have any pressure to stop what they are doing.
Making your husband leave your home is a good way to separate his issues from affecting you and your family directly. He needs time to sort out what he wants to do. At the same time, you need to decide what your husband needs to do in order to earn your trust back and be let back into the family, or if you even want to let him back in. Infidelity and alcoholism are two difficult issues to work through. It is possible, just not easy. And he will have to want to work through them in order for there to be any progress.
In the meanwhile, you may want to consider seeing a counselor to help you decide what you want to do and for support. You need to focus on yourself and how you want to cope with what your husband is doing. To find a counselor, ask your doctor for a referral. Or you can search on line at http://therapists.psychologytoday.com/rms/
You may also want to share your feelings on an on line forum for support. Talking to others who have been through what you are going through can help you a lot.
Self help is another option. Learning more about how you deserve respect in your marriage and how to handle what your husband is doing can help you decide what you want to do about this. Here are some resources to help:http://www.dailystrength.org/c/Infidelity/support-group
Should I Stay or Should I Go?: A Guide to Knowing if Your Relationship Can--and Should--be Saved by Lundy Bancroft and JAC Patrissi
If You Loved Me, You'd Stop! What You Really Need to Know When Your Loved One Drinks Too Much by Lisa Frederiksen
I hope this has helped you,