Ok, that helps. One of two plist files might be corrupt. This isn't always the reason for such things, but it works from time to time, so try this:
Find these files
HD/Users/YourUserName/Library/Preferences/com.apple.desktop.plist
HD/Users/YourUserName/Library/Preferences/com.apple.finder.plist
Move them (try one, then the other if you want to know which one was corrupt, if either one is) to the Desktop. You don't need to have the Desktop Folder open, just drop the file(s) on the background, the actual Desktop.
Restart. This will create a new default plist file to replace the now missing one in that folder.
Test it. If the problem is gone, Trash the file you placed on the Desktop. If not, try the other file, same drill.
How'd that go? (There are more things to try if it's still a problem.)
P.S. I created a new folder in the same location on my Mac and could do both of the things you are unable to do. So there is a fix, we just have to find it.