I've been doing Paralogue 2, and it's really hard to save Anna. I got to her and talked to her once, but she just flirted with Chrom. She didn't actually join my side though and got shot by an archer next turn. I could still get the healing staff from the town even if she was dead, and I hear you can still recruit her even if she dies. Is there any advantage to keeping her alive?
Anna doesn't die so much as retreat. Even if she retreats and the Barbarians take the item from the village, it has no bearing on the next time you meet. That's where it pays to keep her alive (and relatively out of harm's way, if possible).
While she's alive in your current mission, she lowers the difficulty by acting as a fantastic decoy. Keeping her in that state is a much more difficult proposition, but based on the conversation with her next time you meet, your actions in this mission have no effect on her joining or not.
Based on her character page on the Fire Emblem Wiki (warning, lots of spoilers), she won't join until the next time you meet, even if you manage to keep her alive this time.