Bilbo said, talking to himself, "what have i got in my pocket?" (it was the Ring). Gollum thought it was a riddle and Bilbo decided to make it one. he gave Gollum three chances to guess and gollum lost after four. then gollum went "to get something" (it was the ring), and Bilbo thought he was abandoned and left with the ring on. Gollum thought Bilbo knew the way out and he wanted to follow him, realizing Bilbo had the ring. Bilbo saw Gollum racing towards the gate and followed him to safety.
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Because Gollum said that if Bilbo won he would show him the way out of the cave.
Bilbo and Gollum played riddles. If Gollum won he could eat Bilbo. If Bilbo won Gollum had to show Bilbo the way out of the cave.
Gollum promised to show Bilbo the way out if Bilbo won. And Gollum was going to eat Bilbo if he could.
Because Bilbo won it in his game of riddles with Gollum.
Sméagol was Gollum's original name.
He was so busy cleaning up after the party and breakfast that he never got around to his normal dusting.Bilbo had been too busy cleaning up. The dwarves had made their own breakfast and left all the dirty dishes. If he had followed his normal routine, he would have dusted the mantle much earlier in the morning.
At the Unexpected Party, when gandalf introduced the 13 dwarves to Bilbo.
Gollum inadvertently showed him the way out because he thought ''It found the way in, it must know the way out!'' So he ran to the exit in order to catch him, but he was following him the whole time.