Deleting local files DOES NOT delete them from the server. Here is the simplest way to delete files & folders from the server.
In the Files panel, click the Expand button. This will open a screen that shows local files on the right and remote files on the left.
If you do not see a list of remote files, click the Connect button.
To delete a file or folder, simply select it and tap Delete on your keyboard.
View the File Transfer video demonstration at http://cis2.cuyamaca.net/jreed/movies/212/site_management/upload.html .