When was the stove invented?

Answer

In part, the invention of the stove dates back to William Thomson, later Earl Rumford. He was an adventurous man, born in the USA in 1753, with an extensive interest, especially in science. You can read his life story at: http://en.wikipedia.org/wiki/Benjamin_Thompson

In the period 1785-1796 he was a counselor to the prince of Bavaria, whom he helped to modernize his army and to organize relief for the poor. Until then, the soldiers in the field ate what they could get their hands on and cooked something over open fires. Rumford invented the field kitchen: an iron stove on wheels that allowed the preparation of “Rumford soup” (which contained barley, peas, potatoes and stale beer), a nutritious and balanced if not tasty concoction. That stove was already designed to direct the airflow through the wood and to make maximum use of the heat.
Later, in England, he designed a fireplace with a narrowing towards the chimney: this increased the speed of the hot gases and increased the draft, which he could also regulate by means of a valve; this made the room non-smoking. Those principles were then taken over by Benjamin Franklin (the one from the lightning rod) to develop a stove for the living room, in cast iron, with well thought-out airflow and regulation of air supply and smoke extraction.

These were two well-documented inventions at the end of the 18th century.