Yes, there is no order rule in Italian; however, "io e tu" fell out of usage and it's now "io e te", grammatically incorrect but idiomatically accepted by modern grammars (http://www.treccani.it/enciclopedia/io-e-te-o-io-e-tu_(La_grammatica_italiana)/).
While technically grammatically correct, I and she isn't the order in which we would list the pronouns of those eating apples in English. Is this order common usage in Italian? Also, for some strange reason, Duolingo accepted "me and her" as correct, but not "I and she," which is a direct translation of the sentence. The former is grammatically incorrect, so I'm very confused!
"She and I" is "proper" English, i.e. the one people use in formal contexts: "me and her" is "popular" English, i.e. one that is very common in everyday life, e.g. https://en.wikipedia.org/wiki/Me_and_Her (a UK movie). "Her and me" seems to be used too, if less, but "I and she" isn't really part of any dialect as far as I can tell: after all, people who use the "proper" pronouns tend to follow "proper" word order too. Google trends can be useful for some dialect comparisons, e.g. https://trends.google.com/trends/explore?geo=US&q=%22she%20and%20i%22,%22i%20and%20she%22,%22me%20and%20her%22,%22her%20and%20me%22
Even though She and I is the grammatically correct version in English, we are supposed to be translating the sentence from Italian and hence it should be literally translated according to the Italian grammar not according to the English grammar. Duolingo, please accept 'I and She' as the correct answer, as we're literally translating it from Italian.
Not really. It may seem similar in meaning, but that depends on context. Imagine a group of people have gathered. Somebody stops by and says, "what are you eating?" Someone says "He and I are eating bananas." Someone else says "My husband, my son, and I are eating oranges." I say "She and I are eating apples." Someone else, speaking for the group, says "WE are eating fruit."
Because "me" is the object pronoun. Since you're doing the action, you're one of the subjects of the sentence. Hence "I" is correct. If you're ever unsure, just remove everything but the "me" or "I" and see if the sentence makes sense without it. "me am eating apples"? Clearly wrong, so you know to use "I" :)
First, "her" is objective case.. The nominative case is needed, because it's the subject of the sentence: I eat apples. She eats apples. She and I eat apples.
Second, in English, standard polite usage calls for always listing yourself last in a series: "She and I," not "I and she."