I'm looking for a way to automatically sync Dropbox folders to my home server, but in a way that deleting things from Dropbox does NOT delete them from the corresponding folder on MY server.
Can this be done, and how do I do it?
That's not something that Dropbox does. If a folder is syncing it will always be a two-way sync.
You can tell Dropbox to ignore a file or folder, but this will ignore it completely. It won't sync down either. The way to use it in this case would be to let Dropbox sync the folder to your server, then tell Dropbox to ignore the folder so you can delete it in the cloud. This would be a manual process for each folder you want to delete from the cloud.