The dragons do make sense. I think I'm mostly wondering if they have magic powers that we've yet to see. But yeah, I remember there were shooting stars when the dragons were born, and it was this big sign. I wonder if Daenerys will have this magic-amplifying effect in various places she visits along the way. Maybe it's already having an effect in Westeros... maybe this explains Beric Dondarrion being able to come back to life so many times, which (judging by the Red Woman's reaction) seemed like a new thing. Maybe what's being interpreted as the Lord of Light's power originates from Daenerys! That would be sort of mindblowing.
It's hard to tell what the Red Woman's motives are. We'd expect her to be a power-hungry manipulator etc. etc., but this might be another instance of GRRM undercutting our expectations. Her motive might be pure zealotry. She didn't think twice about switching focus to the white walkers. She might simply be moved by whatever she sees in that fire. I think she could turn on Stannis at any time. For example if she hears about the dragon birth and all of that or witnesses Daenerys doing something similar, how could she not view that, at the very least, as a miracle by the Lord of Light?