First, skin color. While dark skin is better protected from ultraviolet radiation, it does absorb more of the energy in visible light than light skin. But most of the sun's heat comes in invisible infrared radiation. Dark and light skin are the same color in that range. Dark skin absorbs no more heat than light skin does.
Hot trays melt the frost and make better thermal contact with the cooling coils.
But there's more. Some of the hot water evaporates in the cold dry air. There's less to freeze, and evaporation speeds cooling. So that story really can be true. Your body has to be warmer than the room to shed heat.