How can there be superconducting protons and neutrons inside an incredibly hot neutron star? It is hard enough for me to try to wrap my head around the idea of superconducting particles other than electrons (especially neutrons!), ....
Given the insanely hot temperatures inside a neutron star or magnetar, several times hotter (at least) than the core of our sun), do the protons and/or neutrons ever actually superconduct?
Have we even created a superconducting material of protons (or neutrons) here on Earth?...
 A: Mature neutron stars (not proto-neutron stars right after supernova explosions) are "relatively" cool if we compare their typical temperatures with the Fermi temperature of their constituents (the Fermi temperature increases with density, that's why is very high in neutron stars, see e.g. the pedagogical notes Neutron Stars for Undergraduates).
Since the Fermi temperature of neutrons is much higher than the actual temperature (exactly like for metals on Earth), an effective zero-temperature treatment is a good approximation in many cases: "the neutrons are degenerate", in the sense that their temperature is low enough that the effect of Fermi statistics is very important.
In this regime of density and temperature, degenerate neutrons can become superfluid via the Cooper pairing mechanism, which is the same mechanism responsible for the "terrestrial" fermionic superfluidity of electrons in superconductors, Helium-3 or other fermionic ultra-cold gases.
Cooper pairing of neutrons (neutron-neutron pairs) and protons (proton-proton pairs) in neutron stars is due to some attractive channels of the nuclear force (see also this and this for the nature of the strong interaction between nucleons).
For more technical details, this is a good review on superconductivity and superfluidity in nuclear systems: Superfluidity in nuclear systems and neutron stars. E.g., here you can find why there are no neutron-proton Cooper pairs and all the details on the attractive channels of the nuclear force at different baryon densities.
A: Superfluidity is essentially always present in nuclei in ordinary matter, which are (almost always) in their ground states. This superfluidity can be reduced by adding large amounts of angular momentum, or by increasing the temperature to the point where it is comparable to the pairing energy of ~1 MeV. ("Temperature" is an approximate concept for finite systems.) Ca. 1970, it was believed that the superfluidity would be fairly easy to totally extinguish, but this doesn't turn out to be the case. The most straightforward way to see that nuclei are superfluid is that their moments of inertia for collective rotation are much less than the rigid-body values.
You need to realize that in units of Kelvin, these are big temperatures we're talking about. 1 MeV/k is about $10^{10}$ K, so for a nucleus, a billion Kelvin is quite cold.
A neutron star is basically a huge nucleus. For some discussion, see: https://academic.oup.com/mnras/article/453/1/671/1751309?login=false

A few hundred years after birth, neutron stars are in thermal
equilibrium and have temperatures of $10^6–10^8$ K (Tsuruta 1998; Page,
Geppert & Weber 2006; Ho, Glampedakis & Andersson 2012). While this is
certainly hot in terms of terrestrial physics, the temperatures lie
well below the Fermi temperature of nuclear matter, which is of the
order of $10^{12}$ K (Sauls 1989).

