Is there a droplet-like application available for the Mac that I can drag a file to, have it uploaded to my server and the URL copied to my clipboard?
Take a look at Automator. It should be in your Utilities folder.
I would write a shell script using OS X’s built-in FTP command, and then use pbcopy to copy the path to the clipboard. Once the script was working, I would use DropScript to turn it into a droplet.