How to delete a managed folder and its contents programmatically via hook_uninstall?
Drupal doesn't consider a folder to be a group of files, and doesn't have a method for bulk deleting managed files (like it does for unmanaged files: file_unmanaged_delete_recursive). So you'll need to identify which files you want to delete manually. A query with WHERE uri LIKE 'public://folder_name/%' might be sufficient in this case, for example, but you'll know you're own setup better than I can guess.

Once you have a list of file IDs, load up the objects, loop through them, and call file_delete on each. If you have the File Entity module installed, you can shorten this by using file_delete_multiple(), and just pass the IDs.

Once you're happy all the files have been deleted (you could stat the folder after running the loop), then remove the folder with drupal_rmdir().

Bear in mind that file_delete() is not guaranteed to remove the file; other parts of the system can declare their interest in a managed file by way of the file_usage table. If you want to unconditionally remove files regardless of any other entity in the system that they're attached to, you can use the file_usage_* methods in file.inc.