What does it mean that the neutral pion is a mixture of quarks? The quark composition of the neutral pion ($\pi^0$) is $\frac{u\bar{u} - d\bar{d}}{\sqrt{2}}$. What does this actually mean?
I think it's bizarre that a particle doesn't have a definite composition. There's a difference of 2 MeV between the quark masses and I don't understand how this can be ignored. If I were to somehow manage to make a bound state of an up and an anti-up quark, what would it be? Would it be a variation on the neutral pion or would it somehow transform into the mixture?
 A: David gives a complete answer on the mechanism. I will tackle:

There's a difference of 2 MeV between the quark masses and I don't understand how this can be ignored. If I were to somehow manage to make a bound state of an up and an anti-up quark, what would it be? Would it be a variation on the neutral pion or would it somehow transform into the mixture?

You seem to imply that assuming the probabilistic interpretation, if the two photons coming from the $\pi^0$ decay are measured, half of the time their effective mass should be smaller because the $u$ quark has a smaller mass than the $d$ quark.
1) Within a bound state particles are virtual. Virtual means that their mass is not constrained to be the invariant mass they would have as a free particle. Think of the nucleons in the nucleus, the proton and the neutron in deuterium.
The mass of a proton is $\ \ \ 938.272013(23) \ \rm MeV/c^2$, 
while that of a neutron is $\ \ \ 939.565378(21)\  \rm MeV/c^2$ 
While a simple sum gives $\sim 1877 \ \rm MeV/c^2$, the mass of deuteron is $1875.612 859 \ \rm MeV/c^2$
The difference is called binding energy, but the point is that neither the proton nor the neutron can have their invariant mass within the bound nucleus, they have a virtual mass. 
2) even though mass and energy are connected through $E=mc^2$, mass is not a conserved quantity in special relativity. 
Going back to vector addition in three dimensions: when operating  a vector addition, the lengths are not conserved. Two vectors may add up to a zero length vector if they are in opposite direction  and of the same magnitude. 
Mass is the equivalent measure in special relativity four vectors, it is the "length" of the four vector and follows vector algebra. It is not conserved. 
The argument: since $d$ quark has a larger invariant mass than a $u$ quark the combination of down  anti-down must  have a larger invariant mass than the up anti-up one is wrong. The four vector algebra comes out that both have the mass of the $\pi^0$.
3) As the quarks are always bound in hadrons, there are two definitions of a quark's mass, the current mass, the one entering the QCD equations which are the ones you quote,  and the constituent mass. That last is the mass with the accompanying gluons within the hadron and is the same for up and down. 
A: This is not "just quantum mechanics", it is more than that. Quantum mechanics tells you that states $u\bar{u}$ and $d\bar{d}$ are allowed to mix, so that you can consider a $u\bar{u}$ system as a sum of $(u\bar{u} + d\bar{d}) + (u\bar{u} - d\bar{d})$, but it doesn't tell you that they have to mix.
If these states didn't mix, and they had approximately the same energy, then there would be no paradox, you would be free to think of the pion as a $u\bar{u}$, or as $d\bar{d}$. If the $u$ and $d$ have a different mass, then $u\bar{u}$ and $d\bar{d}$ would be the right way to visualize the "u-pion" and the "d-pion", even when there are reasonably strong interactions.
But for the actual pions, the symmetric part is split in energy from the antisymmetric part by hundreds of MeV, five times the mass of the pions. This splitting is what makes the pions counterintuitive, and to answer the question you need to address the splitting.
Saying that pions are made of quarks is like saying that sound is made of atoms. It's true that if there are no atoms, there is no sound, but that's about it. The QCD vacuum is like a condensed matter system, and it has a quark condensate at the pion scale. The eigenstates of motion of the quark condensate define the low-lying excitations of QCD, and the lightest motion of the condensate is moving its parts chirally against each other. By this, I mean turning the left-handed u/d and right-handed u/d quarks in the condensate by an opposite phase. This would do nothing to the energy if chiral symmetry were exact, that is, if the quarks were massless. This means that you could "move" the vacuum in the chiral direction without any energy cost, and this gives massless "phonons" (Goldstone bosons) for this process, by moving the vacuum over here a little, and not moving the vacuum over there. These phonons carry the same quantum numbers as the isospin triplet $u\bar{d}$/symmetric/$d\bar{u}$. These phonons are the pions.
The mass of the pions is not zero, but it is small compared to other strongly interacting particles by a lot. This reflects the fact that the up/down quarks are light compared to the QCD scale. While this picture is only accurate to the extent that the pion mass is small (and the pion is not that light), it is indispensable for understanding pion scattering. Because while the pion mass is visible at scales of 7 to 8 fermis, the interactions with stuff like the proton take place at a scale of 1 fermi, where the pion mass is negligible.
The reason pions are split from their isospin zero partner, the eta-prime, is because the gluons in the vacuum already break part of the chiral symmetry by themselves, through instantons. This splits the two kinds of chiral sound, the pion and the eta, and neither of them is made up of quarks like a molecule is made of atoms. The eta-prime vacuum sound mode is five times stiffer than the pion vacuum sound mode.
When doing quark analysis of light mesons, one must always keep in mind that they only tell you the symmetry numbers, the isospin, strangeness (or SU(3)) quantum numbers. It is only at high energies/high masses that quarks become constituents of the hadrons and mesons in the ordinary sense.
A: 
I think it's bizarre that a particle doesn't have a definite composition.

Yeah, it is. As qftme said, that's quantum mechanics for you. It really doesn't make sense until you immerse yourself in the subject for long enough (and even then, only somewhat). But it does appear to be the way the universe works.
Anyway, just so everyone is on the same page, let me start from the basics. If you're familiar with linear algebra, you know that a vector in a 2-dimensional vector space, for example, can be written as a linear combination $\alpha|0\rangle + \beta|1\rangle$ of two basis elements $|0\rangle$ and $|1\rangle$. For example, a direction vector of length 1 that points northeast can be written as
$$\frac{|\text{north}\rangle + |\text{east}\rangle}{\sqrt{2}}$$
or it could be written as
$$|\text{northeast}\rangle$$
or
$$\alpha|\text{north-northeast}\rangle + \beta|\text{east-southeast}\rangle$$
etc. You could figure out what the coefficients $\alpha$ and $\beta$ are in that last case, but it doesn't matter. The point is, there are an infinite number of ways to decompose any vector.
The pion state is an example of such a vector. It's often considered to be a member of a three-dimensional vector space. One possible basis for that vector space is $u\bar{u}$, $d\bar{d}$, and $s\bar{s}$. But another possible basis is
$$\pi^0 = \frac{u\bar{u} - d\bar{d}}{\sqrt{2}}$$
$$\eta = \frac{u\bar{u} + d\bar{d} - 2s\bar{s}}{\sqrt{6}}$$
$$\eta' = \frac{u\bar{u} + d\bar{d} + s\bar{s}}{\sqrt{3}}$$
This basis is useful because these particular combinations happen to be relatively stable; in other words, when a particle consisting of any combination of $u\bar{u}$, $d\bar{d}$, and $s\bar{s}$ is detected in a cloud chamber (if you're old-school) or a calorimeter or something like that, it will behave like one of these three particles. It's possible that what was actually emitted was the quantum state $u\bar{u}$, but in terms of the "stable" states, that is
$$u\bar{u} = \frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{6}}\eta + \frac{1}{\sqrt{3}}\eta'$$
(hopefully I did the math right). So you would have a probability of $\frac{1}{2}$ that it acts like (or technically, collapses to) a pion, $\frac{1}{6}$ that it collapses to an eta meson, and $\frac{1}{3}$ that it collapses to an eta prime meson. One of those three possibilities is what you'd actually observe in your detector.
You can do this the other way around, too: suppose that instead of $u\bar{u}$, you started with a pion, and instead of measuring the "stable" meson type, you were able to directly measure the quark content. Since the pion state contains equal components of $u\bar{u}$ and $d\bar{d}$, your hypothetical quark flavor measurement would give you one of those outcomes with 50% probability each: half the time you'd find that you had an up quark and an anti-up quark, and the other half of the time you'd find a down and anti-down quark. That's what the state $\frac{u\bar{u} - d\bar{d}}{\sqrt{2}}$ actually means: it governs the probabilities that the pion will interact with a quark flavor measurement as each particular quark type.
