No, you do not. Backing up to iCloud is not the same as syncing to iCloud. Furthermore, you do NOT need your laptop to do an iCloud backup with your iPhone. Also, if your contacts, calendars, reminders and notes are associated with an email account like iCloud, you do not need to back those up at all because they are automatically "synced" to your account online. That means that if you change devices, all you have to do is configure that email account on it and allow those items to sync to that device.
12-24-2015 08:56 PM