The common is close to the square so I assume that all of the characters were familiar with it even if they've never mentioned it. The body may have been dumped there just because it was the closest place that would be deserted at the time (to give the killer some breathing space before Lucy was found). Bodies are commonly dumped/buried in woodland.
How the body was moved needs to be explained for any potential killer, but it'd be more difficult for a woman of course.





