What would happen if we fired all nuclear missiles on earth at the sun?

"So once they utilize all their fuel, lack of gravity in outer space will ensure that they are lost in space."

There is no "lack of gravity" in outer space. We are all in a gravity field with contributions from every single piece of mass in the universe.

Provided that the rocket can escape earth's orbit and produce a delta v countering earth's 30 km/s velocity, then if the missile is stationary relative to the sun by then, it will gradually accelerate into the sun.

The Sun will feel a small itch