How is the expansion rate of the universe obtained from Baryon Acoustic Oscillations? I understand that we can measure the sound horizon from the anisotropies in the CMB. Large galaxy surveys today can construct two-point correlation functions, which reveal peaks at distances equal to today's sound horizon, I think because the BAOs at decoupling essentially created dark matter gravitational wells for structure to begin forming in (correct me if I'm wrong). I'm under the impression that the sound horizon is a fixed physical distance, so its measured comoving distance at any redshift should always be the same. Am I right or wrong in this assumption?

I read some information from a few different sources on this topic which confused me. On Wikipedia: "Peaks have been found in the correlation function (the probability that two galaxies will be a certain distance apart) at 100 h−1 Mpc, indicating that this is the size of the sound horizon today, and by comparing this to the sound horizon at the time of decoupling (using the CMB), we can confirm the accelerated expansion of the universe."

But if the sound horizon is always at the same physical scale, how can we use measurements today to determine the rate of expansion? Do the correlation functions give you proper separations, and from that with different cosmological models we can determine what $H_0$ or $H(z)$ value gives a comoving separation equal to the sound horizon? And is this the same as the BAO signal?
I'm under the impression that the sound horizon is a fixed physical
  distance, so its measured comoving distance at any redshift should
  always be the same. Am I right or wrong in this assumption?


You are both right and wrong at the same time. Right in the sense that, yes, in comoving coordinates, the distance is the same at all redshifts. But when we use galaxy catalogues to measure the BAO distance scale, aka the BAO peak, it usually varies with redshift. This variation is very tiny, of the order of a 1-2 h$^{-1}$Mpc. 


  But if the sound horizon is always at the same physical scale, how can
  we use measurements today to determine the rate of expansion? Do the
  correlation functions give you proper separations, and from that with
  different cosmological models we can determine what 0 or ()
  value gives a comoving separation equal to the sound horizon? And is
  this the same as the BAO signal?


Before redshift of galaxies were measured, people were just using the galaxy positions on the sky (Right ascension RA, Declination DEC) to calculate the angular correlation function. You can also see the BAO peak in the angular correlation function. Nowadays, since we have accurate redshift measurements, people calculate the 3d correlation function $\xi(r)$. To get constraints from BAO peaks in a very simple way, we can in brief, do the following. 

To do so, we first need to convert RA, DEC and redshift to X,Y,Z cartesian coordinates. It is at this step, we assume a fiducial cosmological model, as in $\Omega_{M}=0.27, \Omega_{\Lambda}=0.7, h=0.67$ etc. Now, we have 3D information of our galaxies and we can count pairs and get the $\xi(r)$, from which we can calculate the BAO peak. 

People can then use theoretical model predictions for $\xi(r)$, given a specific set of cosmological parameters. It is here that cosmologists vary the set of cosmological parameters, which will give us a specific $\xi(r)$. 
Lets say, we have 10 variations of $\Omega_{M}$, $\Omega_{\Lambda}$ and $h$. This would give us 30 theoretical $\xi(r)$'s, and we can compute the BAO peak for all of them using an empirical fit model. These peaks can now be compared with the observed peak from the data and we can calculate the $\chi^{2} = \frac{(peak_{obs} - peak_{theory})^{2}}{err(peak_{obs})^{2}}$. This can give us constraints on the different cosmological parameters. 

Of course, this is just a basic answer to your question and there are a lot of other processes involved! You can check out this paper which was recently published in arXiv which uses only the BAO peak information to get constraints on cosmological distance measures.