There are different ways of stating conservation of energy and accounting for energy, which can make the issue confusing. One such statement is "the total energy of an isolated system is constant". This is true, and is the simplest way to state conservation of energy. This form of conservation of energy is the earliest taught.
There's another way of stating conservation of energy, "the energy in a region changes by the amount of energy flowing into or out of a region, and energy in adjacent regions changes by the same amount". You could call this local conservation of energy, and is a much stronger statement. It not only tells us that energy is conserved, but it also tells us that energy can't disappear from a region and reappear far away. This is the kind of conservation of energy that Feynman is considering, so he can apply it to systems that aren't isolated.
The definition of a conservative force is one which can be written in the form
$$\vec F = -\nabla U_F$$
for some function of position $U_F$. We then call $U_F$ the potential energy associated to the force $F$.
If you don't yet know calculus, then for all intents and purposes, a conservative force is one which has a potential energy "partner":
$$\text{Near-Earth Gravity:} \ U_g = mgy \iff \vec F_g = -\nabla U_g = -mg \hat y$$
$$\text{Newtonian Gravity:} \ U_G = G\frac{mM}{r} \iff \vec F_G = -\nabla U_G = G\frac{mM}{r^2} \hat r$$
$$\text{Elastic Force/Hooke's Law:} \ U_E = \frac{1}{2}k x^2 \iff \vec F_E = -\nabla U_E = -kx \hat x $$
So on and so forth. The relationship between the force $F$ and its associated potential energy $U_F$ is such that as $F$ does work on a particle (thereby changing its kinetic energy), then $U_F$ changes as well in such a way that the combination
$$E = \frac{1}{2} mv^2 + U_F$$
remains unchanged. Therefore, if you have only conservative forces acting on an object, then you can simply add all of the associated potential energies to the kinetic energy, call the result the "total mechanical energy," and then notice that this quantity is conserved.
A non-conservative force, on the other hand, is one which does not correspond to a potential energy. Frictional forces are non-conservative, but so are forces which you exert with your hands. Such forces are able to change the kinetic energy of a body, but there is no associated potential energy to compensate, so the mechanical energy of the object will typically change as well.
From what I can tell, this is the source of your confusion - you assume that "non-conservative" means "frictional" and this is not true. At this level, the conservative forces are gravitational, elastic, or (possibly) electrostatic; pretty much everything else needs to be treated as non-conservative until you learn more sophisticated ways of keeping track of energy.
Best Answer
The ball alone does not possess gravitational potential energy (GPE). GPE is a property of the ball-earth system. Therefore mechanical energy is conserved for the ball-earth system, not the ball alone.
Correct. The ball increases kinetic energy but no where in the system (the ball alone) is there a corresponding decrease in potential energy (of any kind). Or, to put it another way, the ball acquires kinetic energy because it is not an isolated system, the gravitational force now being considered "outside" the system.
Hope this helps.