Right well my family situation is far from blissful, only the parents role is reversed.
I would take your mum a cup of tea and some chocolate or something like that, just something nice. Sit with her, chat to her, give her a hug. Just give her a bit of normality.
Your sister - I understand that there is no reasoning with a fifteen year old. However there should be reasoning with your dad! Maybe sit down with him when he's in a good mood, and tell him that he upsets you when he shouts at your sister. Maybe suggest that he leaves it to your mum to do the disciplining or something like that. Make sure that you do it in a nice way.
Are you and your sister close? If you are, you could probably talk to her about her behaviour, making sure all of your language says that you are on her side ("i know dad's a pain, but if you just speak nicely to him, he'll be nicer to you"). If you don't get on, you could try talking to her anyway, but maybe bring her something nice home from town first or something like that. Teenagers are never averse to bribery or flattery !