Gramatically, "y" serves as the complement of the sentence while "elle" is the subject. Être verbs like "aller" only agree with the subject and not the complement or object, so "aller" has to be conjugated according to "elle" whether or not the "y" is there.
And like Sitesurf said, "y" is considered neutral even if it replaces a complement that contains a feminine noun like "à la bibliothèque".
"She has left yesterday" is not possible in English. It has nothing to do with whether she's arrived somewhere. "She has left" means the leaving occurred any time up to the present moment, and by adding a limit to how recent it was ("yesterday") you make it impossible to use the perfect tense.
I'm trying to get my head around the placing of words in French (particularly with negatives, but also here). Why does the "y" come before "est"? It sounds very unnatural for my English-speaking ears, and the only logical explanation I can think of is that "aller" and its auxiliary verb must be kept together as a unit - am I somewhere near the truth on that?
If you remember "il y a" and "il n'y a pas", you will see that "elle y va / elle n'y va pas / elle y est / elle n'y est pas /elle y est allée / elle n'y est pas allée" all work alike.
Bonus for you: it also works with "en": "elle s'en va, elle ne s'en va pas, elle s'en est allée, elle ne s'en est pas allée". (s'en aller = to go (away).
"She went yesterday" should be accepted, since it's far more normal to omit "there" in English even when you would need y in French: "Did she go to the cinema? -- Yes, she went yesterday." And omission of "there" in such contexts has been accepted in every other sentence Duo has given me so far in this lesson.