Does a body at 20

Yes, the body will radiate at 20°C. Every entity with a temperature greater than 0K will radiate. In this case, as the room temperature(30°C) is greater than the body temperature, the temperature of the body would not fall further. Since the surrounding is warmer, the rate of absorption of radiation of the body from the surrounding will be comparatively greater than the thermal loss of the body by its own radiation. As a result, the temperature of the body will does not further fall.