Even more on: Kalman filter for position and velocity
There are some problems:


  1-Did I inject the noise at the correct location? Is it appropriate to
  inject it prior to the double integration?


The noise you injected accounts for process noise at the acceleration state, which you seem to assume as constant plus random disturbance. But then your true state generation is wrong.


  2-What's a good real-world noise power level to use in this
  application? I picked a number at random, but I imagine whatever I
  pick has to be considered in my Q or R matrices. I know if I bump the
  noise up to something like 50, the filter output and measurement value
  go wild, but still track each other.


Which noise? Measurement or Process noise? Where is your measurement noise added? On the simulink model or on the script file? 


  3-To get the simulation to work I had to change my H matrix to 1, but
  [1 0] was specified in the example. Does anyone know why?


H =[1 0] is correct if your state $X = [x ~~\dot{x}]^T$ has 2 elements such as position $x$ and velocity $\dot{x}$. Becasue your measuring just the first state $x$, the position.