Is it possible to "add cold" or to "add heat" to systems? 
Amanda just poured herself a cup of hot coffee to get her day started.
  She took her first sip and nearly burned her tongue.  Since she didn't
  have much time to sit and wait for it to cool down, she put an ice
  cube in her coffee and stirred it with a metal spoon.  After a moment,
  she felt the spoon warm up, but when she took another sip, the coffee
  was cooler.  She was pretty sure, the ice added cold to her coffee,
  and the coffee added heat to her spoon.

Would you agree?
 A: 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.
A: Yes. The fluid is in contact with the spoon, and because the former is hotter than the latter, heat is transferred, resulting in the spoon getting warmer. About the ice cube: technically "adding cold" does not make much sense. The ice cube will melt due to heat transfer from the hotter to the colder medium, a process which cools the coffee. Furthermore, heat is transferred from coffee to water, also resulting in a cool-down. 
A: The most important thing about feeling is that its relative and depends on the temperature of sensor point on skin. That's the reason we need thermometer to measure temperature with which everyone could agree.
Energy transfer is involved here, but it has very little to do with Amanda's feeling. Amanda's tongue and mouth was at high temperature due to initial sip which caused her to feel next sip cooler which was really cool due to addition of ice.
In case of spoon, spoon can't get hotter because heat of coffee (and some of heat of spoon) was consumed in melting ice. But, Amanda felt the spoon warmer because her fingers were relatively cold due to ice touch.
A: I definitely disagree.
Cold is a relative feeling, cant be added; 
Coffee will release temperature to equalize the temp.
A: This is not an opinion based exercise.  Thermal energy is generated through conversion.  It is not inherent.  The natural state of empty dark space is absolute zero.  Stars and other celestial bodies generate thermal energy which is distributed through space and absorbed and retained by other objects.  It is not possible to add cold.  It is only possible to add or remove heat.  This is most easily observed in the concept of computer hat sinks.  They cool the cpu by drawing heat into the metal and using air movement to remove it from inside the chassis
