The usual statement of the zeroth law is that if A is in thermal equilibrium with B and C is in thermal equilibrium with B then C is in thermal equilibrium with A i.e. they are all at the same temperature and no heat exchange will occur between them. Yes, this allows us to design thermometers but the obvious implication here is that if material on one side of a heat exchanger is at the same temperature as the material on the other there will be no passage of heat between them.
With respect to the Second Law what we are interested in is the Carnot's statement of it: There is no engine operating between two reservoirs as efficient as a Carnot engine. As the efficiency of a Carnot engine is easy to calculate (1 - Tc/Th) we can bound the performance of a cyclic machine. As a refrigeration machine uses a Carnot cycle run backwards we easily get the maximum COP of a refrigeration machine as Tc/(Th-Tc) = 1/(Th/Tc - 1). Thus the closer Th to Tc the more heat you can transfer per unit of work. This is relevant to discussions of where to set evaporator temperature relative to your "hot reservoir" (air) temperature. Obviously the closer you can get it the higher the machine's COP, SEER etc and the happier the tree huggers are. Obviously (Zeroth Law) you can't have it higher than the lowest temperature you want to cool to.
With respect to the Second Law what we are interested in is the Carnot's statement of it: There is no engine operating between two reservoirs as efficient as a Carnot engine. As the efficiency of a Carnot engine is easy to calculate (1 - Tc/Th) we can bound the performance of a cyclic machine. As a refrigeration machine uses a Carnot cycle run backwards we easily get the maximum COP of a refrigeration machine as Tc/(Th-Tc) = 1/(Th/Tc - 1). Thus the closer Th to Tc the more heat you can transfer per unit of work. This is relevant to discussions of where to set evaporator temperature relative to your "hot reservoir" (air) temperature. Obviously the closer you can get it the higher the machine's COP, SEER etc and the happier the tree huggers are. Obviously (Zeroth Law) you can't have it higher than the lowest temperature you want to cool to.