'The size of an atom' using Uncertainty Principle

The average vector momentum of an electron bound to an atom is exactly zero. (Otherwise, the electron would leave the atom!)

The average magnitude of the momentum can't be zero, because of the uncertainty principle. So Feynman is using the approximation $\vec p = \vec 0 + \Delta p \hat p$, where the magnitude $\Delta p$ comes from the uncertainty principle and the unit vector $\hat p$ points in a completely random direction.

As for your second question, you're almost there. The kinetic energy does become larger for an electron nearer the nucleus — and, thanks to the uncertainty principle, so does the momentum! It has to be this way because the kinetic energy is approximately $T=p^2/2m.$