MacOS – Notes Application not able to open after upgrading to MacOs Sierra

No. They are gone. iCloud, for a lack of a better word is a "sync conduit" between your devices; it's not a backup. This means that a change you make on any of your devices is what gets synced.

When you restored the 1 week old backup from Time Machine, more than likely, Notes did a sync which overwrote what was on iCloud because it "assumed" that was the latest data to be synced.

What you should do in the future....prior to upgrading your OS do a manual Time Machine backup so it saves the latest state. This way you know for a fact that what you restore is your most recent version.

It seems like your Mac is booted into the Recovery partition. Shut it down and try to boot it normally - it may not have upgraded at all. Then try the install again. If you have updates for your question, edit your question to add them and then comment on this answer so I notice.

To answer a few of the other questions you had:

• I would expect an error message to stick around, so you probably didn't miss one.
• I doubt permissions would be an issue as you didn't see an error message.
• If upgrading to Sierra was not supported, you probably wouldn't have been able to start the install.
• You can almost certainly fix this yourself! Also, the Apple Store may not charge you for this, so it could certainly be worth stopping by.