When Arya escapes the Red Keep with Needle, she doesn't have time to find/take the sheath, but in later episodes she has the sword and sheath.
During Tywin and Tyrion's discussion in the tent, just after Tyrion has returned with Bronn and the hill tribes, Tywin's breath is visibly condensing in some shots (most noticeably when the camera is behind his shoulder), but not in others.
Sam and the other knights seem puzzled as to why the bodies of the dead knights they found do not smell and show little sign of decay. The temperature north of the wall appears to be very cold, judging from how many layers the knights have to wear it must be below freezing, which means the bodies would have frozen solid in the early stages of decomposition. A frozen body can remain preserved for decades, even centuries.
Robb tells Catelyn that he rallied 18,000 men; later, he tells the Lannister scout to inform Tywin that "20,000 Northerners marching south". This is not an error: Robb takes advantage of the scout's miscounting (a deviation of 11% is plausible) to mislead the Lannisters; indeed, as revealed in the next episode Baelor (2011), they fall for the trick, expecting 20,000 at the battle on the Green Fork.
When defending Arya from the Lannister henchmen, Syrio Forell handily dispatched the first attacker while the others were all several steps away. This gave him an opportunity to take the first henchman's sword to defend himself with steel, instead of ridiculously using wood. The First Sword of Bravvos would certainly have thought of this. Even though he knew he was a better swordsman than the Lannister men, he was greatly outnumbered and would never fall into the trap of arrogance by choosing to fight with a wooden sword in a contest of life and death.
After Barristan Selmy tosses his sword in front of Joffrey and leaves, the right-most Kingsguard facing him (from Joffrey's view) is unable to sheathe his sword, as the four other Kingsguards do.