Thermodynamics: Relation between different types of energy: My Analogy I have always been confused between the different kind of energies in thermodynamics so today I thought of finally getting it right. Here's my analogy(which is wrong and I need help in improving it, please read it once):   

Let us suppose we take a gas at temperature $T_1&lt;T_{room}$ in a beaker fitted with a piston and conducting walls.    

Its present energy can be represented by the following diagram:

When the temperature increases both the pressure volume energy and internal enrgy increase by fixed amounts, independent of path(being state variables), according to the equation $\Delta(PV)=nR\Delta T$ and $\Delta U=3/2\ nR \Delta T$

As enthalpy is the sum of both these energies, even it is a state variable, and its change is independent of the path followed.    

Now event though $\Delta (PV)$ is a state function and there is a fixed change in its value, there are 2 ways of changing (increasing in this case) it. One by increasing volume and the other by increasing pressure.
The problem is that in the process of increasing pressure-volume energy by increasing volume, the system has to do some work, which means it has to throw out some energy. But we can't let that happen. So we need to provide the system with some extra energy that it can throw out to do work. We haven't increased the energy of the system by doing that as it has come back to the surroundings in the form of work.
There is this fundamental change in energy of the system that is absorbed by the system from the surroundings and there is this extra energy $P\Delta V$ that is also absorbed but then thrown out in the form of work. The total energy absorbed is termed as heat $q$ and the energy thrown out is called work $(-w)$. They depend on the path as it depends on the way we increase the $PV$ energy.   

Problem:
The biggest problem with this analogy is it cannot explain $\Delta U=q+w$. According to my analogy $\Delta U+\Delta (PV)=q+w$. This means pressure-volume energy is a part of internal energy and not something different. But then what would be the significance of enthalpy $H=U+PV$?
Moreover I don't know where to put:
1. gibbs free energy as I don't have an intuitive understanding of it, I only know the mathematical formula and the definition: maximum useful work that the system can do and
2. energy lost due to entropy ($T\Delta S$). 

It would be great if someone could help me improve my understanding of thermodynamics as I am tired of getting confused each time I open this chapter. An approach coinciding with this one (something like a venn diagram) would be appreciated but anything is welcome. Please help, I really need it!
This is not an attempt at a complete answer but I think its relevant to share this great explanation from "An Introduction to Thermal Physics" by Schroeder: