A log burner is a metal box with a large surface area because infra-red heat radiation is emitter from the surface and into the room. The bigger the surface area the more heat radiation is sent out at once into the room. The box is painted matt black because this is the best surface colouration for emitting heat radiation. OK, I know matt black is also the best absorber of heat radiation but it would be a stupid heater that absorbed thermal energy from the room! You have to think about the context.
It is true that wood is a renewable resource, which can be given as an advantage. However, it DOES release carbon dioxide into the air when it is burned. We say that it is CARBON NEUTRAL because the trees use up carbon dioxide doing photosynthesis when they are alive. The carbon becomes part of molecules like starch that we then burn. The carbon in the molecules reacts with oxygen in the air to make carbon dioxide as it burns. It is like a mini Carbon Cycle. We say "carbon neutral" because it takes carbon dioxide out of the air when growing and puts it back when burning. You end up with the same amount that you started with. Fossil fuels like coal, oil and gas have been keeping carbon safe underground for millions of years. When they burn they increase the amount of carbon dioxide in the air NOW so we cannot call them carbon neutral.