Is the nucleus of a carbon atom, for example, as dense as a neutron star? I read that neuton stars also contain protons. Thinking more broadly, are we surrounded by quadrillion of quadrillions of miniature neutron stars (atomic nucei) which are all "protected, shielded, and held stable" by the orbitals of electrons?
It is a very interesting question that allows to point out the differences between a Neutron Star and Nuclei. Although the dedicated article in Wikipedia Neutron Star fully covers the information, it is relevant to summarize here the elements.
Nuclei are essentially different to Neutron Stars and some reasons are:
Different bounding force: while Nuclei are solely bound by nuclear force Neutron Stars are bound principally by Gravitational force.
Different densities: The typical densities of the nuclei that surround us are around $\rho_0 = 0.15\, nucleons/fm^3$, whereas in neutron stars densities range from $0.2 \rho_0$ in the crust to $>2 \rho_0$ in the core. This is very much related to the force responsible for the bounding mentioned before.
Different n/p ratio: While most stable nuclei range from a neutron to proton ratio of 1 to 1.5, and we have been able to produce nuclei with these ratios for many isotopes, none reaches the fraction found in neutron stars, which is protons ~ 10% of neutrons! Again this is due to the gravitational nature of the bounding, rather than nuclear.
Electrons presence: It is very clear in Nuclear Physics that electrons do not coexist with protons and neutrons inside the nuclei. But in Neutron Stars they are an essential part of its composition and their presence explains important features.
Magnetic moment: The difference in origin is very fundamental. While Nuclear magnetic moment is originated by its internal structure, (i.e. interaction of the nucleons spins with their angular momentum) in Neutron Stars the origin is the incredible rotation speeds which are reminders of the angular momentum rotation of the Star whose core originated it.
Finally, and in a way summarizing, comparing Nuclei to Neutron Stars is like comparing a a couple hundred molecules to a glass of water: the properties do not scale with size, but rather give raise to essentially different phenomena and behaviour.
The nuclei of heavy elements (lead, gold, ...) approach the asymptotic density of extended nuclear matter (and therefore the density of neutron stars). The lighter elements do not.
That said, it would be an error to refer to nuclei as "miniature neutron stars" because the binding force and dynamics are different. Nor are nuclei protected, shielded or held in place by electrons: atoms can be easily ionized and their nuclei at least as stable in that form as when in neutral atoms.