How can I set the appearance of the background area in Inkscape so that I can tell the difference between an image with a white background and an image with a transparent background?
Many other image view/editing programs have a checkered background for exactly this reason, but I couldn't find anything like that for Inkscape.
In InkScape 0.48:
File > Document Properties > Page > General > Background
Move R, G, and B to 0. Adjust A (Alpha) to your preference (experiment to set to your liking).
Note that this approach will set the color you choose as the background for the saved/exported image.
In InkScape 0.92:
File > Document Properties > Page > General > Background > Checkerboard background
Since version 0.92 (January 2017) you can instead choose a checkered background. This won't effect the background of the saved image, but will override the background shown in InkScape.