Help me settle an argument. What impact does air circulation have on a room with a wood stove? ?

We have a wood stove room in our basement. The basement has a window that leads to the furnace room. In the Window is a small fan that blows air out of the wood stove room. So here’s my question. We have a box fan that sits at the doorway to the wood stove room, the wood stove room doubles as a bedroom that has a futon and tv I sometimes enjoy spending time in. My mother is insistent that the box fan needs to blow air out of the room but the room maintains a much more comfortable temperature when it is blowing air into it from the rest of the basement (which is a considerable amount cooler). The duel fans both blowing air out of the room from separate sides seems to me to be counterproductive and I know that cold air is denser than hot air and the fan blowing in while the other blows out seems like it would not only help with air circulation to the room, but would help you push the hot air upstairs to the rest of the house. My mother says that air coming in would make the wood in the (fully enclosed except for the vents) wood stove burn quicker. What is the actual science behind how the air, heat and wood stove would respond depending on the direction of air flow from these two fans? 

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