Gibbs free energy of chemical compounds How the Gibbs free energy of a chemical compound is theoretically calculated from scratch? 

When saying that the Gibbs free energy of NaCl is -384.1 kJ/mol, this value is related to the chemical bonds of Na-Cl.

We should be able to calculate this value from the sum of energies in chemical bonds of one mole of NaCl? Right? How?
The value you have reported is the standard Gibbs free energy change of formation, which has a specialized definition. A standard free energy change of formation is the free energy change that occurs when 1 mole of substance is formed by reaction of its constituent elements in their standard states at 1 atmosphere of pressure and 25 degrees Celsius. 

Thus, for sodium chloride:

$$\ce{Na(s) +\frac{1}{2} Cl2(g) -&gt; NaCl(s)}\ \ \ \Delta_f G^\circ =-384.1 \text{ kJ/mol}$$

Now, all we need to be able to do is measure $\Delta G$ for this reaction at standard temperature and pressure.

In practice, we will separately detemrine $\Delta_f H^\circ$ and $\Delta_f S^\circ$ and plug them into $\Delta_f G^\circ = \Delta_f H^\circ -T\Delta_f S^\circ$.

At constant pressure (which we are by definition operating at),  $\Delta H$ is equal to the heat transfer  $\Delta H = q = -C_p\Delta T$, which is equal to the heat capacity of the surrounding (in practice the calorimeter, which is reasonably isolated from the universe) multiplied by the change in temperature.

$\Delta S$ is still tricky to measure. At equilibrium, for a reversible process at constant temperature $\Delta S = \frac{q}{T} = \frac{\Delta H}{T}$. If the process is not reversible or you are not at equilibrium, then... then it is challenging.