Is it possible for heat to flow from body having less heat energy to a body having more heat energy? Is it possible for heat to flow from a body having less heat energy to a body having more heat energy? 
 A: tl;dr: Yes. Specific heat and temperature are different measurements.
The specific heat of a material is a function of temperature that describes how much heat it takes to change the temperature of a material by a specific amount $\frac{dU}{dT}$. Statistical mechanics concerns itself with questions like this. If we integrate the specific heat function from $0K$ to the temperature in question, we find a measure of an object's 'heat energy'.
Temperature, on the other hand, describes the tendency for an object to give up energy. In differential form, $\frac{dU}{dS}$, it's the change in internal energy with respect to entropy. A warmer object with less heat (tungsten, for example) may absolutely be more likely to give up heat than colder object with more heat (water, for example).
As far as the heat pump explanations, I'd encourage you to pull the refrigerator off the wall and touch the radiation coils on the back - they are in fact warm.
A: This is "possible" in a way and it's how technologies like refrigerators and A/C units work.
While there never really is heat energy going specifically from something cold to something hot, the net effect that these technologies employ is moving heat energy from a cooler area to a warmer area. Usually through compressing and re-expanding gasses, the heat is transferred from the cool area to the gas as it is re-expanded and then from the gas to the warm area as it is compressed again.
Strictly speaking, systems always try to reach thermal equilibrium, so heat will never flow from cold to warm in a macroscopic system. The only reason compressed gasses works is that expanding them makes them colder than the cool area (so heat flows in) and compressing them makes them hotter than the warm area (so heat flows out).
A: A body does not have heat energy. Heat energy (or just heat) is the flow of energy between two bodies.
You are probably thinking about temperature, and to answer the question, heat cannot flow from a body with lower temperature to a body with higher temperature spontaneously. 
If by "heat energy" you mean internal energy, then the answer is yes, if the body with lower internal energy has higher temperature.
EDIT: After reading your comment, here's very concisely how a fridge works:
The substance you wish to cool is compressed, hence its temperature increases. Heat flows out of the substance to (now cooler) surroundings. Then the substance is expanded to original volume and its temperature decreased in the process. 
