Does the Church have an "official" position on whether the Adam and Eve events portrayed in Genesis literally happened or not?
My guess (that's all it is) is that the Church's "official" position is that there really was a man, Adam, and a woman, Eve, that they really were "our first parents" in some sense and really were tempted to disobey God, and by succumbing to temptation, got the Plan of Salvation rolling. I did read an article in the Ensign years ago that said something to the effect that certain elements of the story were definitely not to be taken literally, notably the idea that Eve was supposedly made out of one of Adam's ribs.
If one takes the account as an allegory or story, is he or she worthy to hold a temple recommend?
Let's hope so.
I can't think of a single temple recommend question that people who interpreted the story as an allegory would have to answer dishonestly in order to get a recommend. We're certainly not asked, "Do you interpret all of the Bible literally?"
Any sources for your positions?
I'll see if I can find the article, but it was a long, long time ago.