Work energy theorem and gravitational potential energy

On the contrary, the work-energy theorem states that
W_net = W_c +W_nc = KE_final - KE_initial

If gravity alone is the only force doing work, then W_nc = 0, and the equation for that special case boils down to
W_net = W_g = KE_final - KE_initial. Watch signage.

 

Assuming there are no other sources of loss (e.g. sound, temperature change, etc.) which is an idealized but very common situation when working out problems. If we are using your initial problem where only gravitational potential is concerned, then yes, what you gain in kinetic as you fall will be reflected in an equal loss of gravtiational potential (reflected in the change of height).

For question two, there are other ways you can gain energy, as I've mentioned above. Heat, sound, etc. But no, conservation of energy for the system does not strictly apply in this sense, because there are losses from system to surroundings due to friction and other possible changes.

 

Energy can neither be created nor destroyed, only transformed to different types of energy.

Since W_nc = ΔKE + ΔPE, then when only conservative forces, such as gravity and ideal springs, do work,
W_nc = 0, and therefore,

ΔKE + ΔPE = 0.

This is the conservation of mechanical energy equation, which, when only conservative forces act and there are no non-conservative forces acting that do work, tells us that in such a system the energy is always constant...it is neither lost nor gained, but transformed from one type to the other, whether the system is isolated (a closed system) from its surroundings or an open system (un-isolated from its surroundings).

When non-conservative forces, like friction or applied forces, act and do work, then the change in mechanical energy is not 0, and energy is apparently lost or gained to or from its surroundings (open system). It should follow, therefore, knowing that energy cannot be destroyed, that the work done by nonconservative forces can be represented as
-ΔE₀, where ΔE₀ is the other energy (heat, sound, chemical, etc.) transferred out of or into the system. This can be also looked at as a closed system, where
ΔKE + ΔPE + ΔE₀ = 0,
the total conservation of energy equation, where again energy of a system is always a constant, but transformed to different forms.