I'll try answering your two questions simply first.  If these responses are too simple or miss the mark, let me know, and I'll edit the response.

**1) What are the state propagation vector and State Transition Matrix (STM)?**
    
   The state propagation vector is simply the position & velocity at a given time. 
    
The STM is a matrix that captures the sensitivity of the propagation to the initial state.  So, it answers the question "If I change my starting x-coordinate by 5 meters, how much will my final position and velocity change?"

**2) How can I use the STM to improve convergence on new Halo Orbits?**

You can use the STM to achieve faster convergence on new Halo orbits by mapping the change you need at the Y-axis crossing back to the starting state.  (E.g. if you arrive at the crossing with a +2 Z velocity, you can use the STM to compute a different initial state that will have a Z velocity reduced by about 2.  (subject to linearization errors)  A good breakdown of the technique is here:

[http://ccar.colorado.edu/asen5050/projects/projects_2012/dowling/introduction.html][1]

Just look for the section called Single-Shooting Differential Corrector.  

  [1]: http://ccar.colorado.edu/asen5050/projects/projects_2012/dowling/introduction.html