- Without setting height and width on a Canvas (both automatic), does the Canvas take on the dimensions of the BackgroundImage?
Yes (caveat - only if it fits inside the screen)
- Can the height and width on a Canvas be set (automatic or percentage) in such a way that the BackgroundImage maintains its aspect ratio? (I can’t seem to find a way.)
Set to Automatic
- In the absence of that, is it possible to find the dimensions of an item (image) in the Media drawer?
Once image is set to canvas, get the width and height of the canvas, from there you can work out the aspect. (There are also extensions that can provide image dimensions to you. Look for Image and Image Metadata by Taifun)

Output: W= 310 H= 450 Aspect= 0.68889
(Once you have the aspect ratio you can resize to suit (if you need to))