The reason that Alice drank from a bottle that says drink me is because she is so gullible and naive that she will do even what a bottle tells her.
The reason the cat is always smiling is because he knows what kind of trap the Alice is going into and he is helping lead her in the trap.
The cat does not really help her but it because of him that she is moving.
The reason the cook is throwing things is because he is looking for his friends unhappy birthday present to give to him.