Whenever I place an image file that has transparent file information, the coloring of background images changes the document. Below are two screenshots- the first of just a solid filled grey rectangle, the second is directly after I place a layered Photoshop file into the document. Notice how the background fill becomes darker. Is there any way to fix this error from happening?
There are two things that happen here. First, colors can shift depending on the transparency flattener color space. It should be set to match the background image color space.
Second, transparency forces ID to do a more accurate screen rendering. In this case, your choice of using a gray background may be complicating things. ID does not recognize grayscale profiles, but treats grayscale images as if they are on the black plate of the current CMYK working space. This can result in either darker or lighter grays than you expect. The transparency preview is quite similar to waht you would see with Overprint Preview or Seapations Preview active, and in fact the most accurate screen rendition you will get is with overprint preview turned on.