What is the typical reaction time for most pilots?
Its a bit dated but here is a paper published by the Navy on visual reaction time you can find. Keep in mind these are navy pilots, often considered the best of the best and in peek physical condition. Based on the publish year I would suspect all subjects were male as well. 

Here is another study by NATO on reaction times for pilots on low sleep. 

This is a nice study comparing the reaction times of pilots to non pilots, might be of some help. 


  An additional question I have related to this is, how can one design
  an aircraft to make it difficult for pilots to enter PIO's?


This is an interesting question and in some regards could be its own question here. One of the big debates (mainly between Boeing and Airbus) is if systems like this should exist and who has the final say in terms of control deflection. Airbus takes the approach that the computers should prevent the pilot of say pulling back on the stick and stalling the plane. While Boeing generally allows pilots to be the final say and while their systems will warn of a stall you can ultimately pill back the column until the plane stalls. 

In terms of how you would design something like this you would have to limit control input after an approach to oscillation was detected.