My friends and I had a game where we mixed the house cards between us and played a game like that. It was quite hard to choose the cards fairly seeing as the people that picked first had the best advantage. I picked last but I ended up winning due to a strange Loras and Roose combo. 

I was wondering if there are any ways to do all of this fairly. A house card point system or some other way to make the game interesting but not unbalanced. 

This sounds like a good idea, but what about cards that directly mention specific houses(Rently Baratheon, Victarion Greyjoy, exc.)? Although this problem could be sovlved very easilly, what happens in the case of Ser Davos Seaworth? His house card directly refferences Stannis Baratheon in its description. Would the card's holder determine pre-game the card being referenced to? 

We just change Davos' ability to whichever 4 card the house with him had. For eg: The red viper was in Davos' deck and when he was used Davos' ability came into play.

Other cards such as salador and victarion just change to which house they're with.