The diagram shows some particles in solid gold. What would this diagram look like if the particles of gold were in the gas state? The liquid state?

So in the liquid state, the gold particles, the gold atoms, would be slightly less contiguous and well-ordered. You will have to exercise yor own artistic skills to represent this (I certainly don't have any!).

I can think of one material whose density in the liquid state is greater than its density in the solid state. What is it?

Just looking at your diagram again, I think considerable artistic licence has been taken with that picture. If you were packing macroscopic spheres, say marbles, do you think that achieve such layering? How woould the layering more likely appear?

In the gas state, the particles of gold would be more spread out and moving freely with no fixed arrangement. In the liquid state, the particles of gold would still be close together but would have more freedom of movement compared to the solid state, with particles able to flow past each other.