I'm not sure this part has been answered. A nucleon is complex object (an acquaintance who assisted programming lattice QCD likened it to weather forecasting, only worse), consisting of quarks and gluons (mostly). From the perspective of a neutrino, it is mostly empty space. Thus, first order, squeezing a given mass into a smaller volume does not change the interaction cross section.
Given that it takes light years of lead to reach 50% chance of absorbing a neutrino, that is enough to conclude that neutrons stars have a small chance of stopping a neutrino.