Questions about coil voltage and phase shift in the WPT system
Because you are measuring the voltage directly across R and L, the capacitor acts like a short circuit for the square wave drive voltage edges and, those "edges" pass through to R+L unhindered; just like in this simulation of a tuned RLC series circuit that I "threw together": -




  Why is there a phase shift between primary and secondary voltages in
  this system?


If the two coils are not perfectly tuned (or if they are but are close together) you will have a detuning effect that produces significant phase shifting. It's not a problem.