Why don't the relative positions of stars and other objects in a galaxy change over the year? Though the identifiable stars in a constellation (say Andromeda) and a galaxy (say the Andromeda galaxy) are situated light years apart, why doesn't the galaxy appear at different positions with respect to the stars at different times of the year?
 A: The Andromeda galaxy is 2.5 million light years distant, so its parallax is only in the area of micro arc seconds due to the seasonal position of Earth. This is much too small to be noticed.
A: There are a few effects which cause small changes in the apparent position of stars.
Apart from parallax due to the change in position of the Earth around the sun, and so the angle we see stars at, there are effects due to the speed of the Earth and small changes in the Earth's tilt.
The Apparent Position of Fundamental Stars calculates the change for 1000 of the nearest and brightest objects. For more distant objects, and certainly for other Galaxies, the effect is too small to care about
A: You almost answered your own question by noting the tremendous distances between stars. Stars, and galaxies too, are indeed moving relative to each other but the great distances renders that motion nearly imperceptible. It's measurable in astronomical imagery though, and astronomers are indeed aware of it. The changes are too small to be noticeable over a human lifetime, once again because of the great distances involved.
A: Other answers have given the correct answer. I would like to add a few numbers, and offer an illustration you might work through. The distance from Earth to the Andromeda Galaxy is $2.57\times10^6$light years. Over the course of six months, Earth is displaced by the diameter of its orbit, 2 AU, or $3.16\times10^{-5}$ light years. 
You can try drawing all three objects on a sheet of paper as three dots, and connect them with the sides of a triangle. Call the dot representing Earth $E$ and the angle adjacent to it $\epsilon$. The Andromeda Galaxy appears next to the stars in Andromeda in the sky precisely because $\epsilon$ is small, so you should draw it as small. Now move $E$, according to the scale of your drawing, and see how much $\epsilon$ would change. If you keep things to scale, $E$ will move $10^{-11}$ times the length of the line you drew between $E$ and the Andromeda Galaxy; this is probably smaller than the dot you drew for $E$. $\epsilon$ will change by a similarly-sized fraction. That shift in $\epsilon$, as others have noted, is called the parallax.
If you want to go further, try moving the dots representing the Andromeda galaxy and the other object around the paper, and even allow yourself to make $\epsilon$ large. Again, if you keep things to scale, the parallax will be tiny.
A: They are. However, they scales on which they are occur are so large that they are practically indistinguishable to us puny humans on timescales of mere years.
