Can heat(?) flow from a cold object to hot object? When we dip a spoon (stainless steel) into ice cream,  does it becomes cold or stay the same temperature? If it does, can we say that heat(?) can flow from cold to hot objects?

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*Is this the reason that we are given wooden(or plastic) spoons to eat ice cream?

 A: Theoretically, it is possible for heat to flow from a cold object to a warmer object, but the chances of it happening are almost equal to zero. This is (theoretically) possible due to the second law of thermodynamics which says that the entropy in an isolated system tends to increase. But this law is not a fundamental law; it is, rather, a statistical law and belongs to the subject of statistical thermodynamics.
To not get into too much details, the chances are almost 100% that the heat will flow from the spoon to the ice cream. The reason we use wooden spoons instead of metal spoons is due to the fact that heat travels faster from the metal spoon to the ice cream than is the case for the wooden spoon, because metal is a much better heat conductor than wood. The metal spoon will thus cool down more quickly than the wooden spoon.
A: Other answerers have handled the intended case here, but there is actually a sense in which you can get heat flow from a system with lower temperature to a system with higher temperature. Heat will flow from a system with negative temperature to a system with any positive temperature. This is a bit of an trickery of notation- negative-temperature systems are hotter than positive temperature systems. Thermodynamic Beta $1/kT$ can be thought of as "coldness", and negative temperatures are reached by going through zero coldness rather than through zero temperature.
A standard example of a system with negative temperature is in a laser once population inversion is achieved.
A: Heat is transferred from the spoon to the ice cream thereby reducing the internal energy and hence the temperature of the spoon.  A metal spoon is more effective at this heat transfer than a wooden spoon.
To effect heat transfer from a cold to a hot object you need a refrigeration cycle (pump, evaporator, compressor, throttling valve, and appropriate working fluid).
A: Heat is energy transfer due solely to temperature difference. Heat "flows" naturally from higher to lower temperature. In order for heat to flow from low to high temperature (e.g. a refrigerator or air conditioner), work must be done (by a compressor).
Since the initial temperature of the spoon (room temperature or about 25 C) is greater than the temperature of the ice cream (less than 0 C where it begins to melt), heat flows from the spoon to the ice cream, lowering the temperature of the spoon.
The use of plastic or wooden spoons is probably for economical reasons rather than for their thermal properties (lower thermal conductivity than metal).
Hope this helps.
A: Heat can flow from cold to hot objects, as, e.g., it takes place in refrigerators and air-conditioners, but it requires external work on the system. This work is necessarily produced by a "heat engine" (which could be something like a nuclear power station) which works due to heat being transferred from a (another) hot to a (another) cold object. Overall, the heat flows in in the direction favoring equalizing temperatures, i.e., towards thermodynamic equilibrium.
Note also that heat transfer from a cold to a hot object results in reducing entropy of the system, but the heat flow in the heat engine supplying work results in entropy increase, so that the overall entropy is increasing.
A: You have the concept of hot and cold. The spoon does not. You have a temperature that you like. So you tend to model all temperatures relative to that. Thus, hot, cold, and just right.
The spoon doesn't care. When you put a spoon in ice cream they both equalize their temperatures until they are the same. Heat always flows from higher to lower. You can imagine a lack of heat flowing from lower to higher but that's just an idea. There's nothing real behind it.
So while it feels like touching the cold spoon is letting the cold flow into you it's really your heat flowing out.
That's because there really is no such thing as cold. It's all heat. Your atoms are moving a lot, or they aren't. You either have heat or you don't.
The only way to "get cold" from the spoon is if someone else sneezed on it.
A: Your idea of what is "Hot" and what is "Cold" is difficult to quantify, think of it as "Hotter" and "Colder", like the spoon is "Hotter" than the ice cream, and somewhere between them is a "Middle" tempature they will average out at over time.
I believe what you are asking to relates to Entropy which is where the hot and cold items must trend towards the middle tempature. However, if you invest energy into the system, you CAN make the hot side hotter and the cold side colder--this happens with refrigeerators but there is a tricky version closer to what you are asking called "Thermoacoustic cooling".
With Thermoacoustic cooling you play sound waves in a tube at such an wavelength that the energy moves the hot to one side and the cold to the other.  This is essentially what you asked for, the heat from the cool side moves to the hot side.
It cannot be done without energy input though, that would break a lot of laws. In essence this is just a solid-state heat exchanger, nothing magic. Google it for 'cool' videos though.
A: Thermal energy is chaotic. At all times, bits of energy (photons) are flying off of nearly every object (objects at $0$K only receive energy) in every direction. Because of this, every object sends and receives energy from every other object it can "see". When we see warm things cool off and the cool things warm up, we say that heat flows from hot to cold, but it's a generalization that hides the fact that a smaller number of photons also go from cold to hot.
As for your room temperature spoon in the freezing ice cream, metals conduct heat more than wood, so the wooden spoon won't feel so cold on your fingers and it won't melt as much of your ice cream over a similar duration.
A: Imagine 2 snooker balls in a table: Every time you shoot one against the other, the faster moving ball will end up moving slower (even if just a little) and the one moving slower will move faster by a corresponding amount. In other words, for 2 isolated objects, the motion always passes from the one that has more to the one that has less$^*$, even though the opposite is absolutely compatible with energy conservation$^*$$^*$.
At a microscopic level, heat transfer is just the process of transferring motion from some particles to the next, and therefore, just as before, the ones with less motion (colder) will gain momentum (heat up) by taking it from the ones with more (hotter). At equilibrium the average quantity of motion at both objects will match, because if the initially colder object gets hotter the process moves in the opposite direction.
All this said, now the caveat: imagine that instead of just 2 balls you have the full set on the table, each one at its own speed. If you just concentrate yourself into a single slowing moving ball, there is a chance that after being stroked by a faster moving one it strike(s) other(s) at its vicinity and ends up moving slower (or even stop). This means that, (at the vicinity of) that ball speed was reduced (it got colder) after being stroked by a faster moving one (a hotter one). However, after sometime, there is a huge chance that the ball will get stroked again and heat up. What this means is that locally at a (very) small part of the spoon, and for a (very) short time interval, its temperature may be reduced by the colder ice cream. However, at the big scale and for a considerable amount of time the overall process will be the spoon cooling down and the ice cream eating up.
In conclusion hotter objects are just much more likely to get colder in overall than the opposite.

$^*$ This is the second law of thermodynamics.
$^*$$^*$ This is the first law of thermodynamics.
