Does the sun rotate or sit still?
Partially correct. Yes, it is spinning. But it is spinning from the centrifugal force and exchanges of energy present as all the material fell into it's gravity well as it was "passing by". It is a long slow descent spanning billions of years.
Rotation itself has absolutely no effect on gravity. And gravity is not related directly to spin. Gravity is entirely caused by mass warping the space around an object. Here's an OVERSIMPLIFIED example:
Suppose you had an extremely heavy object - planet sized - and you just dropped it somewhere deep in outer space away from the gravity well of the sun. It is sitting completely motionless. Then a huge rock like an asteroid comes passing by very closely. Depending on the angle and speed that it approaches as well as the mass of the rock, it could be captured by the planet's gravity and brought smashing into the planet. Since there is no resistance in space, it would impart a small amount of energy to the planet. Unless it was EXACTLY centered, this would give a small amount of rotational energy to the planet. If the rock came close by but didn't hit, it would still pull with it's own weak gravitational field a small amount. It may be hardly measurable at all, but over billions of years of repeats of the same thing, it will start to spin.
All mass has gravity, so all things affect the movement of everything else within their gravitational field. So it's no surprise that all things have spin.
Now don't confuse this with modern techniques used to estimate the mass of an object by it's spin. This kind of measurement is similar to relative dating techniques. The idea is that all things in our solar system are VERY old and that they have all been influenced by roughly the same gravitational forces and that they are mostly the same age. These estimates are used largely to guess the mass of asteroids and comets. They are just general guesses made when there is no other information available to help determine the mass. These guesses are more accurately formed as the objects pass by other massive objects like planets where the effect of the planets mass on the direction of the object and the speed of the object can lead to a much more accurate predictor of mass.
In no case is an objects spin used in the equations to determine mass. Only the rate at which other objects spin around it. A more massive planet requires that a moon be traveling faster to keep it in orbit. Anything slower would "fall" into the planet. The heavier the planet, the faster the object needs to goto stay in orbit. On Earth, an object needs to be travelling at roughly 18,600 miles per hour to stay in orbit. Slowing down is what causes rockets and shuttles to re-enter the atmosphere. If the planet were more massive, it would need to travel even faster to stay in orbit.
On a larger scale, the sun is not ony spinning, but travelling up and down and around the plane of the galaxy which is at the same time surging outward towards the dark edge of the universe.