The water drain from the sink in the bathroom goes to the grey tank on mine. I have never heard of it draining into the black tank. In the first place the black tanks are usually smaller then the grey. It would fill that tank pretty fast. Not good if they done that.
The heat pumps do not heat the bottom compartments. The heat pumps only heat up until the temperature falls below freezing, then it will switch automaticly to the furnace. The furnace does have big four inch hoses that run over the tanks to heat them and keep everything from freezing.