This case study is amazing. You explained very well all your decisions and I truly believe your design creates a better user experience.
Regarding the check out steps, I would choose the “next” button instead because it’s more intuitive but I also think it is a good idea to show how many steps you have left if it is just a few. Perhaps include that as a small and informative text below or above the button “next”.
However I would only add that info if the check out form is a big one. Otherwise, I don’t see the need.
And again, congratulations ☺ this is an inspiration for me!