Does "iCloud Backup" to sync app data to iCloud?

You would have to ask the developer for the specific app but I definitely recommend that you do turn it on in your manage storage backup options. Also, the backup only saves your app data, not the actual apps. When you restore to an iCloud backup, it restores your app data from the backup, then downloads the actual app from the app store. Finally, you won't see anything about the backup on icloud.com. That only shows some of the data you are syncing with your iCloud account, not the data you are backing up. In fact, you can't view the data stored in the backup at all; the only way to access it is by restoring the backup to your device.

The apps listed under Documents & Data are apps that are storing their data in your iCloud account (called "Documents in the Cloud"). These apps are syncing their data with iCloud so that you can share the data across multiple devices sharing the same iCloud account, such as saving a Pages document on your iPhone, then editing the same document on your iPad. This is separate from your backup. You won't see any evidence that the app data is being backed up, either on your phone or on icloud.com, beyond the fact that you can see that your turned the app to On under Backup Options. If it's turned on there, the app data will be included in your next iCloud backup, and will be restored to your phone should you ever need to restore your iCloud backup.

Check Settings>iCloud>Documents & Data to be sure it's set to On.

Try turning it off, then back on. Also be sure the app actually supports syncing documents & data with iCloud (most don't). Finally, just to clRify what I said earlier, turning the backup option to on for an app had nothing to do with documents & data syncing. They are two different things. One syncs data with iCloud, the other includes app data in the backup.

I'm away right now and can't check but you should check iCloud system status to see if they are having server issues. (Just google it.)