That sounds more like carbon monoxide poisoning from leaving the lid off and allowing the embers to smolder.
It takes a lot (a ton) of burning galvanizing (creates zinc oxide) to to make you sick... I weld galvanized steel all the time, smoke even coming up inside my helmet, I've never been sick. I ain't sayin' it's good for me... but, I'm sayin' what you're describing is classic carbon monoxide poisoning. I don't believe a little scorching of galvanizing can cause what you describe in an open home... I doubt hot ashes and embers can even heat zinc hot enough to create zinc oxide. It requires around 800°F to melt zinc, nearly 1700° to boil it, and a whole lot more to actually burn it.
(I know this from my bullet casting using wheel weights... some of which are zinc.)
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