Continuity mistake: Azog kills Fili with his arm sword, running him through from back to front. In the next moment, when Azog tosses the dying Fili off the ledge, we see the arm sword - and it is shiny, clean and spotless.
Continuity mistake: When the dwarves finally decide to leave Erebor and join the battle, they swing a giant gold pendulum into the blockade they all made earlier in the film, knocking it over to create a causeway, across which they all then run in a battle formation, straight into battle. However, just as they all exit the keep, the pendulum is on its downward swing back outside of the wall. In all following shots (from numerous different camera angles) the gold pendulum simply disappears. This is most notable in the shot taken directly in front of Thorin as the dwarves run towards the camera. The pendulum should be visible in the large hole in the wall behind them, but instead the hole is empty, with no pendulum to be seen.
Continuity mistake: During the battle, when Legolas throws Thorin's sword into the troll above Thorin, he pulls the sword out and it is stained with blood. But seconds later, when Thorin is walking on the ice, the sword is completely clean.
Continuity mistake: The number of dead Orcs on the ice as a result of Legolas' arrows changes from significant to small in one shot from his point of view.
Chosen answer: After the battle the Arkenstone was placed with Thorin in his tomb. It is shown that during the siege, prior to the battle, Thorin sends out a raven which returns shortly before the Dwarves arrive. This raven carried the message to ask for help. The mission to retake the mountain was not a secret among the Dwarves, Thorin had sought an army to help them from the beginning but only the few who went on the journey were willing.
Not quite, the mission to retake the mountain was organized by Gandalf and Thorin, who lived in exile with his family and small group of followers. Dain and other Dwarves did not know, as Gandalf never shared this information with anyone, not even Elrond. It was only after Thorin and Co. Actually arrived in Erebor that Thorin asked the ravens to travel to any Dwarf settlement nearby and gather the armies as the Elves and Men would no doubt come back to the Mountain. The raven was actually an old raven, so he only flew to the Iron Hills to get Dain.