Can the Sun / Earth have a dark matter core? If dark matter interacts with ordinary matter at all, it should most likely occur where ordinary matter is densest. Hence we have papers about neutron stars possibly containing dark matter cores (example).
But if neutron stars can have dark matter cores, white dwarfs, the Sun, or even the Earth can have dark matter cores too - they're just less likely to have such cores. If the Sun / Earth does have such a core dark matter would be much easier to study since they're so nearby. Is there any observational evidence that the Sun / Earth is 100% ordinary matter? If no, what is the observational limit of the Sun / Earth's dark matter fraction? I've seen popular-level articles (example) about such theories, but they're all rather speculative. 
 A: As a partial answer, whether or not Earth might have a dark matter core would seem to depend on the type of dark matter being considered.  There are a lot of hypothetical proposals, and the truth may involve more than just one.  Due to the ambiguity of it, there's not currently a public, compelling argument to interpret any experimental evidence as suggesting or refuting the possibility of a dark matter core.
Here's one line of work that discusses the possibility that dark matter may form a hair-like streams around Earth as opposed to either forming a core-like clump or being unaffected by Earth:

A stream of ordinary matter would not go through Earth and out the other side. But from the point of view of dark matter, Earth is no obstacle.  According to Prézeau's simulations, Earth's gravity would focus and bend the stream of dark matter particles into a narrow, dense hair.
Hairs emerging from planets have both "roots," the densest concentration of dark matter particles in the hair, and "tips," where the hair ends. When particles of a dark matter stream pass through Earth’s core, they focus at the "root" of a hair, where the density of the particles is about a billion times more than average. The root of such a hair should be around 600,000 miles (1 million kilometers) away from the surface, or twice as far as the moon. The stream particles that graze Earth's surface will form the tip of the hair, about twice as far from Earth as the hair’s root.
–"Earth Might Have Hairy Dark Matter", NASA (2015-11-23)

Here's how they picture it looking:
            


If the Sun / Earth does have such a core dark matter would be much easier to study since they're so nearby.

This study's authors seem to agree:

"If we could pinpoint the location of the root of these hairs, we could potentially send a probe there and get a bonanza of data about dark matter," Prézeau said.
–"Earth Might Have Hairy Dark Matter", NASA (2015-11-23)

A: The easiest way for dark matter to become trapped inside another object is if it interacts and loses some kinetic energy. Otherwise it would just gain kinetic energy as it fell into a gravitational potential and then shoot out the other side. To be clear - this answer assumes that the "non-ordinary" dark matter that the question refers to is non-baryonic dark matter consisting of particles, as-yet unknown.
In order to interact we have to suppose some weak interaction of these particles is possible and if so this is going to be most effective when there is a large cross-section and interaction probability.
If the mass of an object is $M$ and its radius is $R$ and if the interaction cross-section is $\sigma$, then the following is illustrative.
The number of nucleons is $\sim M/m_u$. The number density of nucleons is
 $$n \sim \frac{3M}{4\pi R^3 m_u}.$$
The mean free path is $(n\sigma)^{-1}$ and the probability of interaction for a dark matter particle passing through the object will be
$$ p \sim 1 - \exp(-2n\sigma R) \sim 2n\sigma R$$
$$ p \sim  \frac{3 M\sigma}{2\pi R^2 m_u}$$
So for the Earth, putting in some estimates for $M$ and $R$, we have $p \sim 4 \times 10^{37} \sigma$; for the Sun we have $p \sim 10^{39} \sigma$; and for a neutron star with $M = 1.5M_{\odot}$ and $R=10$ km, we have $p \sim 10^{49}\sigma$. 
Of course interaction alone is insufficient, the dark matter particle needs to lose energy and there are also considerations of gravitational focusing, the incoming energy spectrum (including the rest-mass of the particles) and the density of the dark matter and rate at which it might "build up". However, whatever $\sigma$ is (and we know it is small), it is 10 orders of magnitude more likely to interact and get captured inside a neutron star, than the Sun (or Earth). I suppose therefore the argument is that if there were any dark matter captured inside the Earth or Sun, then neutron stars must be full of the the stuff. It makes sense therefore to search for evidence of dark matter inside neutron stars.
Dark matter behaves gravitationally in the same way as ordinary matter, however it does not have the same equation of state as ordinary matter. There would therefore be structural differences (a different mass-radius relation) and also differences in the cooling rates for neutron stars (see this popular article for example). Given that we do not know what is at the core of a neutron star, then we don't know that there is no dark matter there. I will try to ascertain what observational limits do exist.
There have been various theoretical studies that show how the capture of dark matter might affect the structure of the Sun (e.g. Cumberbatch et al. 2010). Capture of dark matter lowers the core temperature and could have a potential effect on helioseismology results and the neutrino flux (especially at particular energies --Lopes & Silk 2010; Garani & Palomares-Ruiz 2017). No such effects have been unambiguously detected.
There also could be a neutrino signature from dark matter self-annihilation and this is a possible route to detecting dark matter trapped inside the Earth. Upper limits have been found from the ICECUBE experiment that are of course consistent with there being no dark matter there, but also consistent with the presence of dark matter with small self-interaction cross-sections (e.g. Kunnen 2015; Aartsen et al. 2017).
A: Newton came up with gravitational physics and the theory fitted all observations of the kinematics of the solar system. Small discrepancies disappear by the introduction of General Relativity, which is very consistent with observations of orbits and the use of GPS. As far as I know there are no open questions in solar kinematics which need part of the measured masses of the planets/sun to need dark matter to be explained.
From the link you provided the large masses of neutron stars, usually larger than 1.4 sun masses, are needed for effects to be seen . It is possible that if a dark matter model fits data of neutron stars , it could be tried on the sun. The masses of planets are already calculated with whatever content exists inside their radius and they do not have variable radii as in the example you link, nor luminosity that could have measurable changes, as in two references in the link.
