Tyrion finds out which of three members of the small council (Pycelle, Varys or Littlefinger) is spying for his sister, by giving each suspect a different unique piece of information, and then seeing which one his sister finds out about. This method, commonly known as "Barium meal test" or "canary trap" (the latter term was coined by Tom Clancy in the novel "Patriot Games"), has been used in real intelligence agencies for decades.
The note that Theon writes to Robb, which is only briefly shown before he burns it, reads: "Robb, I hope this reaches you in time. My father has rejected the offer and plans to attack the North, raiding the shores and taking Deepwood Motte. Mobilize your army and make for the North before it's too late. I'll write again when I can. Theon." The scene was not in the book and was shot much later, as a transition before Theon's baptism. It was filmed on a minimal set, which is why the surroundings are completely dark.
The Children of the Forest, a mysterious non-human race who were reportedly the original inhabitants of Westeros, are mentioned in the show for the first time. In Game of Thrones lore, when the First Men arrived in Westeros, they fought with the Children for dominance over Westeros for several millennia, until a peaceful co-existence was finally established. Both races fought together against the White Walkers and were responsible for the creation of the Wall. Many Children were lost in the struggle, and when the Andals started to invade the continent, the few that still remained finally withdrew to the lands beyond the Wall. At the start of the series, thousands of years have passed without contact with the Children, so most people believe that they have either gone extinct or never existed in the first place.
Cersei tells Tyrion that the Martells loathe them. Indeed, the Martells (and the whole Dornish nation) hold a grudge against the Lannisters for the brutal murder of Elia Martell and her children by Gregor Clegane during the sack of King's Landing. Elia was married to Rhaegar, son of the Mad King, and when he was finally overthrown, the Lannisters reportedly ordered Clegane to wipe out every member of the Targaryen family.
When caught with a prostitute, Pycelle was supposed to be completely naked, save for his maester's chain. However, Julian Glover and his wife objected, fearing that these images would appear on social media. The makers agreed, and allowed Glover to keep on his robes.