Thanks for the detailed response.
I was able to check the whole thread. Unfortunately, QuickBooks Online doesn't provide an automatic assign of class tracking when adding the customer payment under the Unapplied Cash Payment Income account. You'll need to manually open the transaction and assign a class to it. I suggest you send us feedback so we can add this feature in the future. You can click on the gear icon, then choose Feedback.
If you have other concerns, please let me know. I'm always happy to help.