How can she get cold at all when she's capable of breathing fire that can roast a fish she's holding at any time? For that matter, if she's not impervious to temperature, why doesn't she catch fire when she breathes fire on a fish she's holding bare-handed? The flames even go straight past where her hand is.
Furnaces with raging fires burning inside them tend not to get too cold.
Yes, but that assumes a biochemical explanation for the ability to breath fire. If it's magical then there's no need for a scientifically logical explanation.