How do I see my backed up photos

You have to restore the backup. Restoring from a backup erases all data currently on your device. Any data newer than when the backup was made will be erased. A restore is an all or nothing restore. You can't access a backup to see the contents and only restore certain files.

Backup - Restore your device from an iCloud or iTunes backup.    

The backup is separate from syncing photos to iCloud. Backing up won't sync your photos to iCloud.

To sync the photos to iCloud, try  Settings/Your Name/iCloud and slide off Photos syncing. Wait a few minutes and then slide syncing on. Allow time for syncing to occur. The photos sync is extremely slow. It could take several days or more for the photos to sync.  

Last resort - have you tried signing out of iCloud, restarting the device, and then signing back in?

For the backup not completing, see if anything here helps.

Backup couldn’t be completed.   *

Backup couldn’t be completed or you can’t restore.  *

Backup couldn’t be completed or you can’t restore (2).  *

Backup couldn’t be completed or you can’t restore (3).  *

I'm a super-inexpert user of iCloud and have a lot of treasured photos that are * only * on my iPhone. So it makes me super nervous to read: "Restoring from a backup erases all data currently on your device. " I tried backing up after purchasing more data but the iPhone indicated the backup required 4 more hours -- after leaving it all night. The iPhone also indicates: "iPhone not backed up" with no new photos on my iCloud (accessed through MacBook Pro). Is the backup something that I can get help with at an Apple store?

OK, that's helpful! Two follow up questions: 1) Where do I do this -- on my iPhone or iCloud? ("To sync the photos to iCloud, try  Settings/Your Name/iCloud..."; and, 2) If the process can take several days, how do I know when it's done and, is my phone basically unusable until I determine this?

You do it in Settings on your phone.

You can continue to use your phone. The sync takes place in the background. The only way to know it is completely done is to check what is in iCloud/Photos against what is in your phone Photos app.

"Slide off Photos syncing" means slide off "iCloud Photos," correct? If yes, what do I do about "My Photo Stream" and "Shared Albums" - only Shared Albums is on right now, because if I attempt to turn it off, the message indicates turning it off "will delete all shared photos from you iPhone." THX!

You are welcome. Slide off Photos syncing means slide off iCloud. For now, do not change Photo Stream or Shared albums.