Yeah. Unless you're now suggesting that every single person that the dragons go near and do not kill are somehow going to be considered Targaryan, I don't see the Tyrion Targaryan thing at all. Not to mention it would crap all over the emotionally invested hatred Tyrion and Tywin had towards each other. That hatred is better when it's real.
Regarding Davos, his options were...
1) Side with the people he sees as good and honest, even if they are few and facing likely death.
2) Side with the people who just murdered their lord commander who you saw as a good, honest person.
3) Run away.
...no brainer, really, for a guy like Davos. He was only ever going to do anything he could for the good guys to win. As for how easy it was to sway Melisandre, she is having such a mid life crisis at the moment over everything that has gone down, I think she was desperate to convince herself that the gods have not abandoned her. She was scared, too. In the situation they were in, it wasn't great either... I mean, if she didn't do what she could, they were likely going to die regardless.