Many thousands of compressors have been killed by wood stoves. As return air is raised by heated air intake, the compressor pressures increase, increasing stress. When the compressor is off in sub-zero outdoor temperatures, lubrication thickens and refrigerant gathers in the outdoor unit. When the compressor is turned on again in a cold condition, stresses are increased. It is kinder to put a compressor shutoff thermostat in the outdoor unit to stop the compressor and start a compressor heater until the temperature gets above freezing when using a wood stove as backup or alternative heat. It is also kinder to let it lubricate and heat itself by running constantly in freezing conditions.
The return above the stove is a great channel for smoke and fire to get to the upper rooms.