we move on Earth as the planet rotates, Earth moves within its orbit in the Solar system, our Solar system moves within its orbit in the galaxy, and our galaxy moves (we believe) within the universe, though I don't think we know how fast or what direction; generally, everything outside our local group is moving away, though.
Yes, all this movement occurs and is infinitely small compared to the distance to other galaxies.
Our sun moves around the galaxy at about 500,000 mph...and thus so do we.
The things aren't moving apart.
Space itself is expanding.
space is expanding AND things are moving apart.
where does red shift come from if things aren't moving apart?