When Robb is fighting off the bandits, his sword's blade is already bloodied before he's able to cut anyone down though the blade was clean when he removed it from the scabbard.
The Moon Door is opened after Tyrion demands a trial, and then it is opened again, just before the trial by combat.
While pure gold requires 1,064.18 °C (1,947.52 °F) to melt - a temperature not achievable in a simple campfire - many gold alloys melt at considerably lower temperatures. Pure gold is almost never used in jewelry because it is much too soft, so we can assume Khal Drogo's gold is alloyed. Hence the fact that it melts is not a goof. However, it does melt extremely quickly over the course of just a few seconds and that is not realistic. You cannot even melt the equivalent amount of chocolate that fast in a campfire.
Gold glows yellow-orange when in a liquid state. It doesn't have a metallic gold color until it cools.
When Ned is reading 'The Lineages and Histories of the Great Houses of the Seven Kingdoms', he turns to Robert Baratheon's entry, where Robert is described as being 'black of hair, brown of eye'. Robert's eyes are actually blue.
Countering Sansa's statement of Joffrey being a 'golden lion', it sound to some like Arya says that he is a 'Stark'. She actually says that he is a 'stag'.
Robert brags about "making the eight" by referring to the feat as having slept with a girl from each of the Seven Kingdoms, plus the Riverlands. But the Riverlands (in conjunction with the Iron Islands) are one of the Seven Kingdoms.