The 1700's house we lived in when I was a kid had dampers in all 6 fireplaces. No one had lived in it since the Civil War- so the dampers were installed before that. They appeared to be original.
Caves and teepees had no dampers. I think Franklin writes about the value of a damper in his patent on his friend's stove. [around 1750 and not much like the 'Franklin' stove we know of today]
Originally, maybe. Houses don't usually have enough airflow to try that today-- you could open a window in the room.
I would guess mid-1800's-- but that's just a guess.