Actually, the basement is a fine place to put a stove, as the heated air will indeed rise to warm both basement and anything above. Moreover, a woodstove operation can be a bit on the dirty side, given that the wood invariably has dirt on it when stacked, brought inside, and moved into the stove. Cleaning out the ashes periodically also can dirty up the immediate area. The basement floor is better suited to that than your livingroom.
For the heated air to move up through the house, cooler air must be able to return to the basement. That's why leaving the basement door at the top of the stairs open will work better than louvered panels in the door. Warm air rises up the top of the stairwell while cooler air from elsewhere in the house sinks along the floor and down along the stairs.
My son has done much of the heating of his house with a woodstove in the basement, keeping two floors above comfortable, as long as he leaves that door open. For our new house, I installed a small woodstove in the basement, both as backup heating (and primary heat this winter during construction) and because I like a wood fire now and then. The stove is in a finished space in the basement, and the chimney pipe goes straight up through a chase in the corner of a bedroom above, through the attic, and out through the roof. It draws very well.
You might pay attention to routing of the woodstove chimney in any house you are considering. Functionally, having the chimney within the structure until it goes through the roof tends to work better, at least on startup, than one that goes outside close to the stove and then upward. From a fire safety point of view, you might want the inspector or someone from the local fire department to review the chimney installation.
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