Revealing mistake: When the master's gold falls into the water off his boat, several of the gold items can be seen floating, something no gold item should. (00:40:00)
Revealing mistake: When Gandalf shakes Bilbo's hand it should be a lot bigger than his but it is the same. (02:25:26)
Revealing mistake: During the battle of Dale, Bard confronts Alfrid, who is dressed as a woman and is trying to escape with looted gold. During the confrontation a troll climbs over the parapet and is killed by Bard. As the troll falls, Alfrid also falls. When he lands, the ground near his hand ripples like fabric covered in sand.
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.