You are right "e" is the short form of "este", always. But is more informal than the long version, so usually is used in speaking (less in writing).
Be careful, that this is for the 3 person singular of the verb 'to be'. For the second pers. singular "ești" you can't use "e" (ești≠e).
You would put "Nu e o fată." if you were to skip the subject which you can do in Romanian as the verb ending does let you know that the subject is 3rd person singular. However, you would not know if the subject were "he" or "she" and you might want to keep the subject for that information when that is important to know. "He is not a girl." is perhaps said if the clothing and hairstyle is not obvious, but "She is not a girl." could be said if she is already a woman. I think the pronouns for "I" or "you" might be omitted more often. I believe that in the beginning lessons, Duolingo is just making sure that we learn the subject pronouns and later they may be used less.