I suppose you might think of the ice cube "adding cold" to the system, but by convention, we typically discuss these matters in terms of heat. If cold was an inverse of heat, we might say that removing heat from a system is the same as adding cold, but in order to be parallel with the scientific community, it would be wisest to say you are "removing heat."
Recall that two bodies in contact will wish to reach thermal equilibrium. So, by adding a cold ice cube, the higher amount of thermal energy in the hot coffee will begin to "leak" into the ice cube, which has a very low amount of thermal energy. These two systems will attempt to reach a thermal equilibrium, when the water from the ice cube and the coffee have the same amount of thermal energy.
Over time, the coffee will also lose thermal energy to the air as it heats up the immediately surrounding air.
The spoon is a the same story. Since it was initially cool (let us assume room temperature), it will be receiving thermal energy from the coffee (which has more thermal energy) and begin heating up. Assuming this spoon is metal, it will be a nice conductor of this energy, and the whole spoon will likely heat up easily. Due to this increase in thermal energy, the spoon would become hotter.
We could say that by becoming hotter, we "removed cold" from the spoon, but again, that is against general convention. We generally speak in terms of heat.