Is there any way of seeing if someone in a shared folder has downloaded or edited a file?
Nope.
There's no way to see whether a person has downloaded a file.
When you call https://www.dropbox.com/developers/core/docs#metadata, the modifier field will tell you who last modified each file in the shared folder. If you call https://www.dropbox.com/developers/core/docs#revisions on an individual file in the shared folder, the same modifier field will tell you who made each edit to the file.
modifier
So no for "downloaded," but yes for "edited."
