Signing out of iCloud on any device, discontinues all related iCloud sync to & from that device.
Calendars and Reminders sync uses iCloud sync and requires being logged into iCloud on the particular devices, where you want syncing of Calendars and Reminders, using your Apple ID and the associated PW. Best way to initiate a correct sync (after logging out of iCloud on all devices - having been careful in following Apple's instructions), is
first logging into your iCloud Account on your desktop Mac(s), using the iCloud Sys Pref, and check the particular Apple Services (say, Calendars and Reminders - recommend checking Contacts sync - and, unchecking the sync services you don't want active). Then, go to each iOS device where sync is to be active and in Settings -> iCloud, log into iCloud and activate the sync for the same Services (say, Calendars, Reminders, Contacts). A subsequent sync will then occur on the iOS device.
On your iOS devices, recommend using Find My iPhone and Find My iPad, both of which require signing into iCloud on that device; but since you're signing into iCloud to sync Calendars & Reminders (& Contacts), just activate these "Find My" services.
You can also use the iCloud settings on iOS devices (with iOS 10.3.x) to see the devices associated with your Apple ID (or, log into your Apple Account via the web URL to Manage Apple ID). There's benefit in confirming that your Apple ID is correctly associating current devices, particularly if you've sold or simply taken devices out of service.
Here's a very helpful link for
iMore Guides Time Machine is an Apple - macOS utility app for backups, setup in Time Machine Sys Pref. Time Machine and SuperDuper have nothing to do with iCloud when backups are being completed using either app.
IMHO, iCloud is a very good, as well as reliable, sync service. I have used it for years without any sync problems. Yes, iCloud had previous issues, particularly after the transition from MobileMe; but over time and especially with recent macOS and iOS updates, mainly macOS 10.12.x & iOS 10.3.x, iCloud has been dependable when correctly setup.
Sometimes iCloud requests the Apple ID & password, and many find entering these details to be a nuisance. However, these entry requests (as well as 2-step or 2-factor responding) are security measures Apple has initiated to insure the user of an Apple ID is the bone fide user. As long as one inputs the correct ID and password, this is only a momentary delay. ...I find it understandable and reassuring Apple has taken steps, given the aggressiveness of data thieves, that assist us in protecting our Apple ID and related, iCloud / Apple Services.
