Interpretation of the interior Schwarzschild solution Consider the interior Schwarzschild metric were the value $r_g$ of the $r$-coordinate at the body's surface is equal to the Schwarzschild radius $r_s$. In this case, the interior solution simplifies to the following form
$$ds^{2} =
\left(1-\frac{r^2}{r_g^2} \right) c^2 dt^2 - \left( 1-\frac{r^2}{r_g^2} \right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right),$$ 
where the factor $1/4$ in $g_{00}$ of the original solution has been absorbed into the time-variable (the correponding time-discontinuity across the boundary is not relevant for the following). 
Obviously, the above expression is corresponding to a static representation of the de Sitter spacetime with positive and constant curvature $K=1/r_g^2$. From textbooks of the standard model of cosmology we know that such a space has the property to act repulsive in the interior of the horizon. Wouldn't this imply that the equilibrium state of the interior fluid is perfectly homogeneous at the end, instead of collapsing into the center $r=0$?   
 A: The Schwarzschild interior solution is a non-physical model that that gives the correct qualitative prediction for the interior metric of astrophysical bodies only within a certain parameter range. It is constructed by assuming that the density of the material of the "Schwarzschild star" is constant and the pressure profile is assumed to automatically enforce this constancy of density. 
Such a fluid could be imagined to be realized, for instance, by a non-conducting ideal gas with the right temperature profile $T(r)$ such that $P = \rho k_{\rm B}T/m$ where $m$ is the mass of the gas particles and $\rho$ the constant mass density. Omitting the issue of conduction and various convective instabilities, this temperature must be at least finite. In the wikipedia article you link to you can find in the section Pressure and stability that the pressure (or temperature) profile diverges at the center when the ratio of the radius of the "Schwarzschild star" reaches $9/8$ times its Schwarzschild radius. Obviously, there is no physical material that can reach infinite pressures and temperatures, so considering a star radius equal to the Schwarzschild radius makes no sense.
To consider more physical equilibria, you can integrate the Tolman-Oppenheimer-Volkov equations, and consider perturbation theory of such equilibria (treated in detail in Misner, Thorne and Wheeler's Gravitation). Ultimately, the most convincing argument that squeezing a physical material up to the formation of a horizon always leads to a singularity is given by the Hawking-Penrose singularity theorems.
A: The interior Schwarzschild metric models a fluid in equilibrium, and it is clearly static. It doesn't represent a collapsing body; for that you need to take a look at, for example, the Oppenheimer-Snyder solution, which models a ball (or was it a shell?) of presureless dust collapsing to a black hole.
A: If there was a material that could resist collapse so that $r_g=r_s$ were possible it would mean that this material would have a divergent bulk modulus. The bulk modulus
$$
B=-V\frac{dP}{dV}
$$
for an isentropic process is $PV^\gamma=k$ a constant and so $B=\gamma P$. However, if the collapse stops at the horizon it means there is a large outward acceleration at the horizon. The near horizon spacetime for the Schwarzschild metric defines an acceleration $g=c^2/d$ for d the distance from the event horizon. This means the acceleration required to oppose the gravitational implosion is divergent and the bulk modulus of this material must be similarly divergent. As a result, this harte kugel state means this metric for the interior of a ball of material at the Schwarzschild radius breaks down. However, the idea may not be completely dead as I explain below.
The Schwarzschild metric for the interior of a material ball
$$
ds^2=\frac{1}{4}\left(3\sqrt{1-\frac{r_s}{r_g}}-\sqrt{1-\frac{r^2r_s}{r_g^3}}\right)^2dt^2-\left(1-\frac{r^2r_s}{r_g^3}\right)^{-1} -r^2(d\theta^2+sin^2\theta d\phi^2),
$$ 
with the condition $r_g<r<<r_s$ has 
$$
ds^2=\left[\frac{r_s}{r_g}\left(9-\frac{6r}{r_g}\right)dt^2 – dr^2\right]+\frac{r_s}{r_g}\left(\left(\frac{r}{r_g}\right)^2dr^2~-~\left(\frac{r_g}{r}\right)^2dt^2\right)-r^2(d\theta^2+sin^2\theta d\phi^2)
$$
which is the metric for $AdS_2\times\mathbb S^2$ perturbed by a space $(g,{\cal M})$. The metric terms for the square brackets are $(g,{\cal M})$ and  terms in the parentheses are for $AdS_2$. It is not hard to show $AdS_2\times\mathbb S^2$ is the near horizon condition for the Kerr or Reisner-Nordstrom metric. Hence the anti-de Sitter spacetime emerges from black hole metrics as well. This condition occurs for a highly accelerated observer positioned above the event horizon. The near singularity condition is what obtains for an observer close to the horizon. This means that an observer very close to a Kerr horizon on an accelerated frame observes a similar spacetime as that near a Schwarzschild singularity. The difference though is that with the Schwarzschild singularity the sphere has a radius that has a dimension given by a timelike dimension. With $AdS$ spacetimes there are also closed timelike curves. In the near horizon Kerr metric by contrast this condition obtains for an accelerated frame, but the $\mathbb S^2$ is parameterized by a spatial radius. There is then some funny relationship between these anti-de Sitter spacetimes and with the de Sitter spacetime..
The $dS$ spacetime above might sound completely fictional. However, there is this oddity with the so-called firewall. This stems from the fact that Hawking radiation emitted by a black hole is entangled with the black hole. Hence this is a bipartite entanglement. As the black hole emits radiation the emitted radiation there is a growing probability this is entangled with previously emitted radiation and with the black hole. Once the black hole emits half its mass, at the Page time of around ${\frac{7}{8}}^{th}s$ the duration of a black hole, this probability approaches unity. This means previously emitted radiation transforms from a bipartite entanglement to a tripartite entanglement. This runs into troubles because the $W$, constructed from bipartite states, and $GHZ$ states are defined on different parts of the $3$-Kirwan polytope for these states. This means there is a $3$-tangle that is a sort of topological obstruction to this sort of unitary transformation. For some more details see  V. Coffman, J. Kundu, W. K. Wooters, “Distributed Entanglement,” Phys. Rev. $\bf A~61$:052306 (2000). If one presumes that quantum states or equivalently quantum information is conserved, then the fix is to abandon the equivalence principle. Other people, Unruh and Wald etc., prefer to abandon conservation of quantum information and preserve the equivalence principle. The firewall then might connect with this harte kugel.
Consider the black hole that has reached the Page time and has a firewall. This is a barrier or naked singularity that is for all purposes the material with a divergent bulk modulus. At least on a classical level that is what it appears as. This would then tend to support some idea of the black hole interior as a de Sitter spacetime. The $dS$ and $AdS$ spacetimes are separated from each other in a higher dimension spacetime by a light cone. The $dS$ and $AdS$ spacetimes are connected by their conformal infinite regions or “boundaries.” The $dS$ spacetime is the single hyperboloid outside the cone and there are two $AdS$ spacetimes are inside the opening of the two conical horns. This then may lead to a nest of questions about these relationships, the near horizon condition, near singularity condition and the firewall.
