Does the earth have a gravitational pull on the sun?
yes. the earth as well as the other celestial bodies, all have their "pull" on the sun and each other.
Yes, there is a gravitational pull that works both ways. Just as the moon is held in orbit by the gravitational pull of the Earth, likewise the gravitational pull of the moon has its effect on the Earth, most commonly seen in its effects on the tides of the ocean. No doubt the gravitational pull of the Sun also effects ocean tides and other unseen effects.
charon has a gravitational pull that is equal to which earth has a gravitational field on charon. still the charon has a gravitational pull that can pull earth a centimeter distance away from the earth yearly but we all studied about law of conservation of center of mass you might have understood why i have taken this topic earth is not moving away from the sun...
The Earth orbits the sun because the Sun has a larger mass than the Earth giving the sun a greater gravitational pull. It is also the same for the moon. The moon orbits the Earth because the Earth has a greater mass giving it greater gravitational pull than the moon. ( I hope you understood that :) )