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No heat needed? Maybe a heat source (like a small 7W incandescent lamp) could provide better results. Fleas are attracted mainly by heat (people and pets do not emit visible light, just IR) :)
HI! this is a cool work! I would recommend search a bit on the net about "rocket stoves", if you want to have a more efficient energy conversion. You will find that the rocket stove has more than 90% of efficiency, thus allowing to consume less wood (or have more heat). An "open" fire like yours generates lots of smoke and wastes most of the wood stored energy.
Usually refrigerator manufacturers recommend leave a few space ~50 millimeters or more on all sides of the equipment, to allow air to flow; however, if one does not decides to fill ALL space around the refrigerator, things should not post a problem. But is good to pay attention in case a particular refrigerator niche is relying on all the space around it in order to function correctly. In practice heat goes up, so upper space is the most important anyway. I have seen below some comment(s) and noted by experience that in fact, region around the refrigerator do get moist and hot. And as such, the wood would need a good paint/finish to avoid a quick dirt and fungus buildup, and in the spices not hermetically sealed as well.
The RRRRRRRRRRBBA, a $3 Arduino