First off i agree with nelco and Taz, you should get support for yourself too, it will do no harm to introduce yourself to the other sections outlined below and there is no shame in it. Drinking does affect all the close family members and what you went through with your father is a huge thing too. Anyway i hope you do that:-)
Hmmm on the question on what to do when your mother comes out of rehab, well there is not a thing you can do to ensure that she does not drink again, that is all down to her. Even if you drove her to AA meetings, read out from the Big Book, tried to get a sponsor for her etc...that would be her passing all responsibility for her recovery onto you so that if she fails than you can share the blame. So i would do nothing whatsoever save make a cup of tea when you normally would and let her get on with it, and be there as you normally would.
You are also not there as an emotional dump for her, this you will, i think, hear a bit more about in the other sections on SR. That is what a counselor is for, see the problem is if i choose to dump all my **** on you, where do you go to dump yours and, now, mine?
If you don't find the information in the other sections (i'm certain you will find many people to help you here) then, if you have the number of the treatment centre, you could ring and ask to speak to one of the staff and ask them the same questions that you asked below:-)
Plese go and check out the other sections, and keep posting, they will have real life experience of what you are going through rather than from us alcoholics point of view.
Your mother is a very lucky lady to have you, that's for sure!