I have a small outdoor ceiling mount light intended for use on a patio ceiling that I would like to use on the wall next to my front door. There is currently a wall mounted fixture there. Can I build a box to attach to the wall over the electrical connections and then mount the ceiling mount light on the underside of the box without running into any trouble?

You can but I can't picture how this will look. Why not just buy a wall mount fixture made for this application?