Moon is in orbit around the Sun (as well as in orbit around the Earth). Another way of looking at it is that the center of gravity of the Earth-Moon system is revolving around the Sun, and the Earth and the Moon are revolving around their common center of gravity. The Sun is pulling on the Moon, but the Earth is "falling" right along with it, so they stay together.
Earth just sits there and rotates. It doesn't move. So the moon and the rock are two objects, but the rock is smaller and has no gravitational force.