"An another" doesn't work in English. "Another" is itself a word that means "an other."
Both "next city" and "the next city" are accepted. If that's exactly what you answered and it was rejected, it may have been a Duo hiccup over which the team has no control.
I was led to believe the e with the diacritic meant that the vowel was simply elongated/dipthongized, but with this audio (espeeecially in slow audio) it sounds like the consonant N is in between the two portions of vowel. Am I right to assume that's in error?
That is not an error. The orthography and pronunciation is explained in our Tips and notes and at many other places on the internet https://en.wikipedia.org/wiki/Czech_orthography#Letter_%C4%9A