1st: The sun "Wobbles" as a result of the pull of the planets and asteroids that orbit it. Because the mass of the planets and asteroids is so tiny with respect to the mass of the sun, this wobble is very small. In fact, the center of mass for the solar system (the point around which the sun wobbles) is actually within the sun. This means that the size of the wobbles are smaller than the size of the sun.
2nd: The solar system moves around the center of the galaxy (as you said) at around 250 km/s. This is due to the fact that our galaxy rotates like a pinwheel. We are near the edge of the pinwheel so we move pretty quickly.
3rd: Our galaxy is moving in the universe. We are currently moving towards the Andromeda galaxy. Movement on this scale is very difficult to measure, so values for these speeds are not well known.
With regards to spin, the sun DOES spin! Because it is a gas, the spinning of the sun is not like the spinning of the earth. The sun spins in regions and at different rates in those different regions. The various regions of the sun swirl and bubble, further confusing the overall rotation. Nonetheless, the sun DOES rotate.
Answered by: David Consiglio, B.S., High School Chemistry and Physics Teacher
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'Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.'