Piezo transducer preamp and AGC/compressor
It uses two 12V batteries. Could I use just one yet be able to give
  both + and - voltages?


The AD743 will work down to supplies as low as +/- 5 volts so try using a mid-rail generator like this: -



All the ground points in your circuit go to the mid-rail point. Choose an op-amp that is low noise.


  Gain is set with R2 and R3. Could I replace both of them with a
  potentiometer to have variable gain?


Replace R1 and R3 with just a single potentiometer - the junction of R1 and R3 is where the wiper goes.


  The high pass frequency is set with R3 and C1. Could I replace the R3
  with a potentiometer and add a switch to disable the filter when it
  isn't needed?


I would just choose a bigger value for C1; after all you don't need it to work down to 0.000001 Hz do you!


  Both AD743 and AD797 are suggested in the diagram.


Read the data sheets. Swings and roundabouts. Look at the noise graphs in the data sheets. It has nothing to do with distortion.


  How would you add either AGC or a compressor?


I would start by justifying that some non-linear process was justified so, over to you.