First, the mistake that Fili and Kili make is that the cave in which they seek shelter when a storm approaches has a passage that opens in the back and through the passage, goblins capture the dwarves.
Second, Gollum's birthday present, as he calls it, is the ring - the ring the renders the wearer invisible. Bilbo stumbles upon the ring when he is in Gollum's lair and he isn't aware, at first, of the ring's power.
Third, the goblins and the Wargs meet before they join forces in the battle of 5 armies. They are in opposition to the dwarves, the men and the elves.