I believe that the COM permission is to the user and not to a specific folder, so, going with your line of thought, there should be a process that writes or deletes a new file to the website?
correct the web application runs in an IIS worker process and executes in the context of the user that is the identity on the IIS application pool that is configured for the web site in IIS
System.UnauthorizedAccessException: Access is denied. (Exception from HRESULT: 0x80070005
definitely looks to me like a file system permission error for the sql ce database file that lives in the /App_Data folder beneath the root of the site. So the user that is the identity on the IIS application pool has insufficient permission on the database file, it needs full control of this file
moving the sql ce file form another machine is a common way the file system permissions get messed up but regardless of how it got that way I do believe it is a file system permission problem.