"There is no other choice."

1) You have to use the existential ann. Basically, this is used when saying something exists, like An bhfuil Dia ann? for 'Is there a God"'

2) rogha eile by itself would be 'another choice' so Níl rogha eile ann would be more akin to 'There isn't another choice', and, while they're synonyms, Duolingo likes the literal translations.

Níl rogha eile ann is “There isn’t another choice”, while Níl aon rogha eile ann is “There is no other choice”. Their meanings are slightly different, though they’re often used synonymously; the former addresses whether another choice exists, while the latter addresses whether any other choice is viable.

Yes, it is. It implies the existence of aon rogha eile