Is the vowel harmony here irregular, or is there a good way to understand why
it is hatOdik and nyolcAdik?
Unfortunatelly there is no rule when O or A.
why is nem vagyok a huszonhatodik... not accepted?
You're contrasting 26th with 28th. You should put huszonhatodik into the focus. The focus is the spot right before the verb. Your translation would be fine if there wasn't a second part to the sentence.
So when you are comparing two things like 26th and 28th vagyok goes at the end of the sentence as if the sentence were affirmative instead of a negation?
Your answer below explains why we don't say nem vagyok which was what confused me. The emphasis is on not being one thing but being something else.
Wtf? The second "a" would not move from its place in the choices pile.