What compact-camera features help expose a black dog correctly in snow or changing light?

The problem with the photo is that the image is under exposed. The camera's meter tries to make everything an 18% gray and, well, that snow looks about an 18% gray to me so the meter was working 'correctly'.

There are a couple different options to look for with this. The first is to find a camera that has a snow or beach mode. An example of what this can do will can be found on Adorma's tutorials. You'll need to look at the scene settings for a particular camera to see if it has something appropriate for that.

Another option is, instead of letting the camera do the work, get one with a bit more advanced manual controls which allow for exposure compensation. Playing with a set of cameras available on DPReview and their camera feature search - compact with full manual and exposure compensation +/-2. These cameras would allow you to be able to specify the exact settings you are after to properly expose the snow.

When taking the photograph, you will likely want to turn on the histogram viewing, if it is available. This would allow you to see if the image is blown out or not... or if it is possible to capture the scene properly (too much dynamic range for the camera).

If worst comes to worst, expose for the dog (spot meter... maybe get an 18% gray collar you can meter against). You can always correct the snow in post production, but details lost in the dog are lost - and thats what you really want to photograph.