Does a reflection leave a "shadow" on an imperfect mirror? Obviously, if I watch a mirror while I walk down a hallway, the image in it changes based on my reference point. Conversely, if I watch a shadow while I walk down a hallway, the shadow doesn't move regardless of where I am (unless it's my shadow). I'm curious if an imperfect mirror (like your average glass door or window) behaves both ways. 
Consider this very beautiful and expertly drawn image (The black here is walls, while the gray is a slightly opaque window):
Assume I am at point A. An observer at point B would just see me through the window, and an observer at point C would see my partial reflection. 
However, would an observer at point D be able to see an image of me projected onto the glass? If the window's opaque at all, then I assume it absorbs light. Would that create a fixed "shadow" which is visible on the window, regardless of where an observer stands? If so, would it be in color? 
 A: 
However, would an observer at point D be able to see an image of me? If the window's opaque at all, then I assume it absorbs light. Would that create a fixed "shadow" which is visible on the window, regardless of where an observer stands? If so, would it be in color?

There will be a shadow — or less inaccurately, a silhouette — on the glass, yes. It will not be a color image of you (or rather, it will be an absence of the color of the sunlight). The image that is "on" the glass, ignoring ordinary transmission, will be the average color of the surrounding illumination just like any matte reflective object, but hard to see because transmission is so much brighter. The silhouette is a particularly easy case since it has the strong contrast between direct sunlight and blocked sunlight.
In real life, the best way to notice this would be if the window is dusty or, for a more extreme case, made of frosted glass, so that there is more scattering of the light than ordinary clean window glass.

You can experiment with the effects of imperfect reflections using a flashlight and a wall mirror. This is not greatly similar to what you asked since there is no transmission through the reflective object, but the geometry involved is related except some things are in the virtual reflected space "behind" the mirror.
If you face a mirror and shine a flashlight at it, holding the flashlight and its aim away from your eyes, you can see several spots of light each corresponding to a different reflection path:


*

*The spot where the main beam is reflected to, on the wall behind you, which you most conveniently see with the aid of the mirror (but that's just normal use of a mirror).

*The spot on the mirror due to imperfections and dust on the mirror, which will be in the same location as if the mirror were not a mirror.

*The bright lens of the flashlight itself, since it does not focus light into a perfect beam and has some all-around spill.


All three of these spots do not move as long as the flashlight is held still, if you don't count ordinary changes of perspective  them through the mirror.
Here is a diagram. The red circles are the apparent locations of the three spots of light produced by the single flashlight beam (not counting the simple case of looking directly at the flashlight).

Note that they are all along the line of aim of the virtual flashlight in the mirror (except the path looking directly at the flashlight, which I did not draw) — this is because all of these are points where the flashlight's light was scattered away from the beam — just like if you used the flashlight in smoke/fog the entire beam path would be lit up.
A: Well, D cannot see you, but D can see your shadow, on the assumption that your shadow is longer than the distance between you and the window.  Some of the shadow will we reflected, but some shadow will still pass through and D can see your shadow faintly.
