Can you extract energy from "hot" things without a temperature differential? I've been reading about extracting energy from heat, particularly Sterling engines. There's always a temperature gradient. Heat flows from the hot side to the cold side through the working fluid. As the temperature differential approaches zero the engine stops.
My question is that why do we even need a temperature differential? Even more broadly why do we use energy to cool things? Let's take a gallon of water at 50 deg C. Water has a specific heat of 4.18JdegC/g. Ignoring the heat of fusion and following Q = CMdT, just by virtue of it being "warm" (323 deg C above absolute zero) wouldn't the water "contain" 3780 * 4.18 * 323 = 5104 kJ of energy (3780 grams per gallon of water). Why couldn't you extract any of this energy as work and as a result of extracting the energy the water cools itself (it gives up energy, for every 3780*4.18 J extracted it should lower by 1 deg C). I'm invisioning the heated water (or anything heated above abs zero) as an energy sink or battery. Theoretically you should be able to extract 5104 kJ of energy from this 50 deg C gallon of water.
In the Sterling engine we could lower the cold side with liquid nitrogen but it takes energy to make this. In an AC system we spend energy to lower the temperature in the room. Since "cold" is the absence of "heat" or energy, shouldn't cooling things actually create energy for us (the hot things that earlier had energy imparted into them give up their energy to make work). It seems wasteful to just waste the heat on warming up something at a lower temperature differential such as the cool side of the Sterling engine or the ambient air by the condenser.
 A: The Ideal
The main problem is that all current technologies seem to waste a certain amount of energy on actually displacing some of the heat energy from A to B. If there were no such inefficiency, then we would in a sense obtain "free" energy as we could displace and focus an arbitrarily large amount of energy from a material into a finite space, and then use the laws of thermodynamics to power (e.g.) a stirling engine. If only there exists a structure that can naturally, without energy input, make one side cold and another warm. However, this gives us a contradiction.
The Contradiction
Due to the second law of thermodynamics and the conservation of energy, this becomes impossible. As having a perfect "structure" that can transfer heat from A to B would violate these laws. It is required that energy be put into the system to actually perform the displacement. Therefore, even if said system is 100% efficient, energy is still put into the system such that no "free" energy is ever gained. 
For reference (2nd Law of Th.) (wikipedia)

When two initially isolated systems in separate but nearby regions of
  space, each in thermodynamic equilibrium with itself but not
  necessarily with each other, are then allowed to interact, they will
  eventually reach a mutual thermodynamic equilibrium. The sum of the
  entropies of the initially isolated systems is less than or equal to
  the total entropy of the final combination. Equality occurs just when
  the two original systems have all their respective intensive variables
  (temperature, pressure) equal; then the final system also has the same
  values.

A Law
Unfortunately, the question is hard to answer:

My question is that why do we even need a temperature differential?

The reason is that the contrary is made impossible by a law. A law in physics and mathematics doesn't necessarily equate a theorem or theory. It only states "such is so." whilst a theory would actually explain "why such is so".
A: A)
Without a temperature differential, there's no "hot". To say something is hot would mean it has a higher temperature than what you're comparing it with. eg. coconut oil, chocolate or gallium will melt in your hand because you are too hot for it to stay solid, but phosphorus or soap will stay solid because you're too cold for it.
Since 'hotness' is a measure of temperature differential, extracting energy from a point will change the temperature at that point. Also to extract (heat) energy, would mean to move it from a higher temperature to lower temperature (potential), just like any other form of energy, whether electricity or fluid dynamics.
In your case of "5104 kJ of energy", 100% of it could be extracted for work if the environment was at absolute zero and your extractor efficiency was at 100%.
B)
Re: "shouldn't cooling things actually create energy for us"
To cool anything, means to do some work, in this case 'cool'. Doing work, needs energy. So to cool anything will need energy, not create energy. As in A) above (the 1st part of your question), extracting (heat) energy from a point will cool it (reduce temperature), which requires work. So the energy potential you would create at that point by this extraction would be equal to the work you've done to extract the energy from that point, with 100% efficiency.
Further reading keywords: entropy, thermal equilibrium, conservation of momentum, conservation of energy, theory of relativity
A: It may be possible using cold fraction from a vortex tube. Use heat source to boil water; use vortex tube to separate the steam into hotter steam and cooler steam; use hotter steam to replace original heat source. So the original body of water produces some kinetic energy and loses a lot of thermal energy. You would need a temporary external heat source but would gain that energy back and then some if the chain reaction consumed enough water.
