When I was a wee snot, during the war years, (1941-'45) I spent a lot of time at my grandparents house. I still remember the old kerosene cook stove in their kitchen. It served well until after the war ended and new gas or electric stoves could be had. The odor of kerosene could be detected until food cooking smells took over. Some really great meals came off that stove. I remember a barrel for kerosene, secured with a padlock in the back yard on a wood rack under the old walnut tree. I would suppose kerosene was rationed like gasoline, and had to be bought with ration coupons. Granny's stove was green, much like the one pictured.
I can't get a good picture of that stove in the picture. It looks alien to me. Our olden kerosene stove is a rectangular steel with a burner in the middle and a stand of about 3 feet. There is a bin or fuel tank for the kerosene and a little oblong metal ball which has the pump. Before lighting up the burner, you have to pump the metal ball to compress the air. That compression is what pushes the kerosene to the burner so the distribution of the fuel is regulated. The fire is good with the blue color and the hissing sound. And when the stove turns quiet that means there is no more compressed air so you have to pump again otherwise the fire will turn yellow and the pot will be sooty. From what I remember, that was our stove for quite a time, maybe until I was 4 or 5 years old. What followed was an electric stove that is not messy and quite comfortable to use. The LPG stove came around when I was in college, I think.
I never heard of a kerosene stove actually. I know kerosene heaters are used frequently, but stoves are a different thing. I think it is one of the items that had its day in the sun many years ago. I would think they could be quite dangerous, but maybe not more than other forms of cooking. I think these are elements that had their time in the sun.