Can different Apple ID co-exist for iTunes and iCloud backup?
The description is not very clear, but yes, you can have different Apple IDs for iCloud and for iTunes purchases (if that's what you are asking).

To do this: on the iPhone, go to Settings > iCloud > E*nter the Apple ID you want to use for iCloud Backup* > Turn On "iCloud Backup".

Now, to setup a different Apple ID for iTunes purchases, go to Settings > "App Store" or "iTunes &amp; App Stores" > Enter the other Apple ID to be used for purchases.

That way, you can sport 2 separate Apple IDs: one for iCloud backup and one for iTunes purchases.