The /App_Data and /Data folders need to be completely writable by the user that is the identity on the application pool so it can create folders and files. Permissions must be configured to propagate to all child objects so that newly created objects have the same permissions.
The index files will be created at /Data/Sites/[SiteID]/index if the permissions are correct, if not it will fail.
I am not able to reproduce the problem with the duplicated child folder, sometimes things like that may happen if you are using the IIS url-rwriter module you may have some rule there that causes the problem.
Hope it helps,
Joe