Neutron star core understanding From what is given here free neutrons (neutrons outside of the atomic nuclei) are unstable and decay in about 15 minutes into proton, electron and an anti-neutrino (in most cases). 
Also given that neutron stars exist, it would be the case that it is gravity that packs the neutrons close enough to remain stable.
As per my understanding, towards the center of the neutron star (the core), the net gravitational force should decrease (the mass at the middle and outer-core of the neutron star contributes to the gravitational force towards the surface however net force experienced due to gravity at the center itself would be zero given that the masses on the rest of the sphere would pull symmetrically at the core resulting in a net zero gravitational force).
With the above understanding, that there is nearly no gravitational pull at the center, the neutrons there would then be free to decay. Is this understanding correct or did I miss something fundamental? (articles linked do mention that there would be matter in other states but not for the above mentioned reason - rather the justification is due to the higher density - but my point is the exact opposite - that the density at the core would be less due to net gravitational force being zero)
 A: Gravity is in balance with a pressure gradient, not the pressure. The equation of hydrostatic equilibrium is
$$\frac{dP}{dr}=-\rho g\ ,$$
where $\rho$ and $g$ are the local density and gravity respectively.
You are correct that $g=0$ exactly at the centre of a neutron star. This also means that the pressure gradient is zero at the centre of the neutron star, which means the pressure and therefore the density are at a maximum.
What stops the neutrons decaying is not the local gravitational field, it is their extremely high number densities and the presence of a small fraction of degenerate protons and electrons.
Consider a thought experiment where you were able to confine a dense gas of pure neutrons. There would be an initial decay phase producing some protons and electrons. But the densities of these fermion  species would build up until they were also degenerate. When the electron Fermi energy reaches the maximum possible from the neutron beta decay process, then further neutron decay is blocked. All lower electron energy states are already filled.
A: To summarize the understanding, although the pressure caused by gravity is the greatest at the surface and decreases progressively as we move towards the center, the pressure still adds up as we move towards the center (progressively increases). As pressure increases so does density until the point (near the center) wherein due to the prevailing conditions, results in the formation of degenerate gas. 
The misunderstanding was to not account for the shell theorem and therefore concluding that there was a gravitational force acting away from the center as a result of the surrounding mass which also negated the pressure.
