Amazon does offer higher cashback on first credit card payment. and speed of payment probably depends upon the card provider. for me stanchart has always taken 2-3 days. but from my understanding, it would be applicable from the date of payment even if you get confirmation later.
I don't like using Amazon for utility payments anyway since in the bank statements, It shows up as "Amazon <something> <something>" instead of showing me the utility/service name. This just makes it difficult when I am auditing everything later in ynab. :/