Are the neutrons in a neutron star on orbitals as in atomic nuclei? Does the internal structure of a neutron star resemble anything like that of an atomic nucleus? I.e., are the neutrons arranged in a shell like structure with different energy levels, and without a distinct location for each neutron?
I know that the force holding a neutron star is gravity, compared to the strong interaction inside atomic nuclei, but does this matter?
 A: Neutron star properties indeed are determined by quantum effects, without a distinct location for individual neutrons. This special form of matter is referred as neutron-degenerate matter. And in a certain sense neutron star can be treated like a giant nucleus. Neutrons in it could be seen as filling consecutive states according to Pauli principle, one neutron for each state up to a certain maximal energy level.
But the number of neutrons is so large that speaking of individual levels, shells etc. (which is appropriate for the systems of few particles) is pointless. 
However there are some differences: Neutron star is not in a ground state like a stable nucleus (and not in a distinct excited state like say nuclear isomer) but is rather a thermal system, with temperature and thermodynamical properties with Fermi-Dirac statistics. 
Additionally neutron star has outer regions (actually usually having more volume than volume of mostly neutron matter) which have a lot of 'life' other than just neutrons: electrons, various atomic nuclei, ions, magnetic fields... 

the force holding a neutron star is gravity...

Indeed the neutron star is the result of balance between the gravity which tries to pull the matter together and the pressure of degeneracy opposing it. And because the gravity force acting on a single nucleon is so much weaker we end up with such a large object.
