So I 'm building a Baratheon deck but I got a Lannister and Stark demo deck alongside my Core box. I wanted to know the playability of these cards. Am I allowed to use the plot cards contain within the demo decks? What about the other cards. Thanks.

All the cards in the Stark and Lannister demo decks are in the core set Stark and Lannister decks as well. And yes, it's perfeclty alright to run cards from another house in a deck, though keep in mind taht you're paying a 2 gold penalty to do so. Also, the standard deck is 60 cards and 7 plot cards, and you'll probably need more than just 1 core set to build a proper deck.

I was mainly looking to use the plot cards but all good nonetheless. Thanks.