Above is correct. As long as you made 12 payments you are eligible for an upgrade. So your 12th payment would be taken care of on 8/27/18. In my case, I did the upgrade on 9/21/17, I checked my eligibility and my 12th payment will be made on 8/21/18 and therefore come September, I will be eligible for an upgrade.
As for your last question, typically if you made 11 payments and you upgrade, you will end up paying the last payment at check out when you upgrade to the new phone. I started the iUP program when I got the 7 Plus but started in October. My 12th payment would be in September and would be eligible for the upgrade in October, but since they allow you to upgrade early, they just charge you whatever remainder of the 12 you have left at check out so in my case my 12th payment was taken care of at check out when I upgraded to the 8 Plus. Hope this helps.