This gets complex. Personally, I have an oven which has burners on the top (a stove) and the part below where you bake and roast things called an oven. Some people have separate units: a stove top located in one place and an oven or ovens located in another place. The stove top might be on a counter top and the oven(s) could be located in a wall. This is most typically for a more expensive house, in my opinion. I hope this helps.
Yes, it depends on context: in a house the furnace belongs to the heating system, for example in a basement or garage. Instead, the oven is in the kitchen and that's where "we cook".
I see several comments wondering why "cooker" is not a substitute for stove. I live in Virginia, and have never heard that term in all my travels across this country. Perhaps it is an England-English or Canadian-English expression, or some other regional term; but it is definitely not widely used in the US.
In American English when speaking of home appliances we usually say there is a stove in the kitchen. It is a kitchen stove. There is usually, but not always today an oven in the stove and on the top of the stove we would have burners. So is it correct that el horno is the oven part of la estufa, which is the whole appliance?