Doubts on Work-Energy theorem for a system

[Soren4]

While studying energy conservation on Morin I found this explanation about the work-energy theorem for a system.

The work–energy theorem stated before is relevant to one particle. What if we are dealing with the work done on a system that is composed of various parts? The general work–energy theorem states that the work done on a system by external forces equals the change in energy of the system. This energy may come in the form of (1) overall kinetic energy, (2) internal potential energy, or (3) internal kinetic energy (heat falls into this category, because it’s simply the random motion of molecules). So we can write the general work–energy theorem as
$$W_\textrm{external} = \Delta K +\Delta V +\Delta K_\textrm{internal}.$$ For a point particle, there is no internal structure, so we have only the first of the three terms on the right-hand side.

Using Koenig theorem $$\Delta K_\textrm{system}=\Delta K +\Delta K_\textrm{internal}$$ so we have

$$W_\textrm{external} = \Delta K_\textrm{system} +\Delta V$$

I've got two main question on that:

1. Why are only external forces considered for the work?
2. How is the formula above related to the following? $$W_{conservative}=-\Delta V$$

Here are my consideration/doubts:

1. Considering a system of $n$ material points the following holds.

$$\sum W=\Delta K_\textrm{system}$$
But here $$\sum W=\sum W_{i}=\sum \left(W_{i}^{(\textrm{ext})}+W_{i}^{(\textrm{int})}\right)$$
The amount of work considered is the sum of the work done on each point (both from external and internal forces).
And in general we do not have that $$\sum W_{i}^{(\textrm{int})}=0$$
Counterexample: two masses attracting each other gravitationally.

2. If we use the formula reported above we have $$W_{external}+W_{conservative}=\Delta K$$
But does this make sense?

 

[drvrm]

Why are only external forces considered for the work?

as the external forces are doing work -its being considered.