I'm not an expert in German, but I'm quite sure "a woman eats an egg" should be accepted as a translation. I would say that is the most straightforward translation. If it's not being accepted, I'd say someone at Duolingo has broken the translation table for German and English. In fact, at the top of this discussion page, I see the translation being given as exactly that, Translation: A woman eats an egg. Are you sure that's what you entered?
slytherclaw is right,
"Das Ei" is neuter for some reason, maybe because you don't know what gender chicken comes out of an egg: male or female? :) so Egg is neuter in German. But
"Die Orange" is feminin to till its bones. In accusative, only masculine nouns become
"ein + en = einen",
"der + en = den".
'Einen' is the accusative masculine article. 'Ei' is a neuter noun, and so uses the accusative neuter article ('ein', the same as in nominative). 'Apfel' is a masculine noun, and so it's nominative article, 'ein', because 'einen'. Only the masculine article changes between nominative and accusative.
The situation is a little more complicated than this, and can be best described with a table: http://en.wikipedia.org/wiki/German_articles . By the way, in accusative, "einen Apfel" (m), "ein Ei" (n), "eine Zeitung" (f). As for "an", that's an English word (of course, you shall properly translate your German sentences into English). But while in English, "an" vs "a" is dependent on the beginning of the next word, in German there are other things to take into account, the case and the genre of the words that follows the article.
In formal German, there is no difference; thus, you get away with a saying like this: "Man ist, was man isst" / "You are what you eat," which I find much more amusing in German.
In more dialectal/informal German, though, I'm reading that the "t" in "ist" can be dropped, while the "t" in "isst" won't be, so you'd be able to tell the difference in a conversation potentially. Elsewhere too, you might run into something like "isch" for "ist" and "issd" for "isst," which would also set it apart.
Otherwise, I'd say you would have to trust by context. "A woman is an egg" is essentially nonsense, while "A woman eats an egg" makes much more sense. Homophones exist in English too, and we don't struggle with them because of ingrained context. It doesn't help, perhaps, that Duolingo is somewhat famous for its nonsense sentences.
No. Both mean a/an/one, depending on context and english rules ( an before a vocalic sound, a everywhere else). The difference is that ein is for masculine and neuter nouns and eine is for feminine and plural nouns. Also, you're going to learn other variations that are used for different grammatical functions.