Why is the second stellar structure equation first-order ODE? the 2nd structure equation is first order, but we seem to have two boundary conditions (e.g., $dP/dr = 0$ in a star’s center and $P=0$ at the surface) – but first-order ordinary differential equations can only have one boundary condition! What’s the resolution to this conundrum?
 A: The condition $P=0$ at the surface is not an imposed boundary condition, because you do not know the radius of the star.  The procedure for solving the equation (as you have it set up) is based on beginning at the center of the star and integrating outwards.*  Once you have integrated out to the first point where $P=0$, you have found the outer radius of the star.
*To do this in practice, you need to supplement the first-order differential equation for $P$ with an equation of state, which makes the practical problem into one of solving a second-order differential equation.  However, that is not actually related to the issue you are asking about.
A: The equation of hydrostatic equilibrium requires that the pressure gradient is zero at the centre. It isn't a boundary condition.
$$\frac{dP}{dr} = -G\frac{m\rho}{r^2},$$
where $\rho$ is the density at $r$ and $m$ is the mass enclosed within $r$.
The latter varies approximately as $r^3$ and so the RHS tends to zero as $r \rightarrow 0$.
As an analogy, I could say $dy/dx = x$. Then $dy/dx = 0$ when $x=0$, but that doesn't help obtain a particular solution; you need the value of $y$ at some particular $x$. 
In this case you are demanding that $P=0$ (or equivalently $\rho=0$) at the radius that encloses all the mass of the star. There are 4 stellar structure first order ODEs in total, usually expressed in terms of the enclosed mass, so 3 more boundary conditions are required. These are usually the temperature at a radius enclosing all the stellar mass and that the enclosed mass and luminosity flowing through $r$ are both zero at $r=0$. Solving the equations with these boundary conditions then gives the central density and temperature and the surface radius and luminosity.
