The Earth is rotating, but it's easy to compensate for that so that the stars don't turn into streaks. You just mount the telescope and camera on a motorized mount that turns slowly to counteract the rotation of the Earth. That's cheap and easy to do, so everyone does it unless they specifically want to show off the rotation of the Earth.
Every star in that image has some proper motion, usually dozens or hundreds of km/s, and there's no way to counteract that. For any single exposure that's not large enough velocity to turn the stars into a streak, but it is enough for the stars to be in different positions relative to each other from year to year. Since different panels of the image were captured in different years, the overlaps between those panels won't match perfectly. When stitching the panels together, the photographer will probably have edited the photograph so that each star is taken from only one panel. That will prevent any stars from being cut in half or blurred together from both panels, but there could be stars that moved fast enough to have crossed the boundary between the panels and they could show up in the full image twice. Good luck spotting them though.
The Earth is also moving around the Sun, so there is an additional apparent motion, but it's smaller in magnitude to than the proper motion and won't cause any extra problems.
The gas clouds are also moving. Some of them are expanding, some of them are being pushed by stellar winds, etc. Stars can form in dense clouds, and then the stellar wind from the new star pushes all the remaining dust away to create a bubble. You can see these motions over the course of years, so presumably they caused some minor mismatches between panels that were photographed at different times. Since these features are fuzzy, it's not hard to blend neighboring panels together even when they're taken years apart.
https://www.youtube.com/watch?v=ofCooIkIwvQ is a great explanation with good visuals, and it has photographs taken by the New Horizons probe that directly show stellar parallax when compared with photographs taken at the same time from Earth. Even that is only visible on the nearest stars; none of the other stars in those photographs were close enough to move at all.
The Earth is rotating, but it's easy to compensate for that so that the stars don't turn into streaks. You just mount the telescope and camera on a motorized mount that turns slowly to counteract the rotation of the Earth. That's cheap and easy to do, so everyone does it unless they specifically want to show off the rotation of the Earth.
Every star in that image has some proper motion, usually dozens or hundreds of km/s, and there's no way to counteract that. For any single exposure that's not large enough velocity to turn the stars into a streak, but it is enough for the stars to be in different positions relative to each other from year to year. Since different panels of the image were captured in different years, the overlaps between those panels won't match perfectly. When stitching the panels together, the photographer will probably have edited the photograph so that each star is taken from only one panel. That will prevent any stars from being cut in half or blurred together from both panels, but there could be stars that moved fast enough to have crossed the boundary between the panels and they could show up in the full image twice. Good luck spotting them though.
The Earth is also moving around the Sun, so there is an additional apparent motion, but it's smaller in magnitude to than the proper motion and won't cause any extra problems.
The gas clouds are also moving. Some of them are expanding, some of them are being pushed by stellar winds, etc. Stars can form in dense clouds, and then the stellar wind from the new star pushes all the remaining dust away to create a bubble. You can see these motions over the course of years, so presumably they caused some minor mismatches between panels that were photographed at different times. Since these features are fuzzy, it's not hard to blend neighboring panels together even when they're taken years apart.
https://www.youtube.com/watch?v=ofCooIkIwvQ is a great explanation with good visuals, and it has photographs taken by the New Horizons probe that directly show stellar parallax when compared with photographs taken at the same time from Earth. Even that is only visible on the nearest stars; none of the other stars in those photographs were close enough to move at all.