Do the stars move in the sky? Why?
The rotation of the Earth produces apparent movement, as does the yearly orbit.
There is also Proper motion. which is also from an Earthly viewpoint, but too small to be seen without detailed measurement. But was correctly suspected because some stars were not where Greek sky-charts had placed them.
Our sun and all the other stars have quite fast movements in their orbits round the galaxy. Tens of kilometres per second. But not easy to spot, because the distances are gigantic.
Stars with similar motions are assumed to have a common origin. Mostly as a dispersing Open Cluster.
Yes, all objects in the universe move. However, stars are so far away that it takes years of precise measurements to calculate their speeds.
A simpler method is to measure the shift in their light emissions. If they are moving towards us, their spectra are shifted toward blue, and those moving away are shifted toward red. Those moving laterally require long term positional observations, since their light is not shifted.
Yes, they are actually moving to some degree. It's just that the distance is so great they appear to be static. Plus, if you believe the universe is still expanding then you must believe everything is moving, which they are.