Unless there was major gravitation interference by maybe another galaxy or black hole, no, Or if Earth was to change its axis, which would also be another gravitational interference. It all depends on the situation.
Yes, the galaxy is moving. Astronomers feel it is moving in concert with other galaxies (as in a cluster) towards a formation that astronomers refer to as the "Great wall". It also may be moving towards the Andromeda Galaxy however we cannot differentiate between our actual motion and Andromeda's. We are approaching each other, as a blue shift in light has been measured. However the idea that we are stationary, and it is moving towards us is the most mentally satisfying way of describing it. The distances though between us and it, or the great wall are so large however, that it is really of no concern for us for millions of years to come, but we are in motion.