She is the overcomplicated story-arc for the next year or two that will be resolved with a mess of fanwank and/or dismissed with a TARDIS-load of nonsensical timey-wimey bullshit. I can overlook such things though, because she is goddamn adorable.
I think this is about right. I suspect it's just the year's plot (the only one that spanned more than a year is River and that's because she didn't appear every episode).
Anyway, I suspect the answer is none of the above and involves some kind of splitting in time. My guess is modern day Clara is first and whatever happens happens later. Basically, they are echoes in time created by something. But I'm not really going to think about it, I'm fine just enjoying the ride.
As for Clara dying in each episode. It's an intriguing idea, but would get old quickly. The episodes aren't long enough to have to introduce her as a companion each time. And, while she may have some vague notion of herself in other timelines, she doesn't seem to remember everything.