What are the experience requirements and process for becoming an FAA Designated Pilot Examiner?
You can start with the FAA's reference page for Designated Pilot Examiner.  

It will point you to the FAA Order 8900.2 General Aviation Airman Designee Handbook.  It's all there.  You'll need to talk to your local Flight Standards District Office (FSDO) as they are responsible for administering the program. The general eligibility requirements are:  


  
  General Eligibility Requirements. The managing office must carefully consider all of the requirements before designating an
  applicant or renewing a designee. The general requirements for
  appointment, which are reviewed by the NEB, are:       
  
  
  a.    Minimum Age. Minimum age of 23 years for all designees.      
  
  b.    Personality Traits. Must exhibit integrity, professionalism, and
  sound judgment.   
  
  c.    Training Requirement. Must have satisfactorily completed the
  applicable training requirement before appointment/renewal as a
  designee.      
  
  d.    Knowledge of Guidance Materials. Must have access to, and
  adequate knowledge of, the current regulations, policies, directives,
  and related guidance material appropriate to authorized functions.    
  
  e.    Command of English Language. Must be able to fluently read,
  write, speak, and understand the English language.     
  
  f.    Objectivity. Must maintain the highest degree of objectivity
  while performing authorized functions.     
  
  g.    Good Record. Must have a good record in the aviation industry.  
  
  h.    Technology Access. Must have internet access, as well as access
  to a computer and printer with hardware/software sufficient to support
  the required activities.     
  
  Note:  A designee is not required to be a U.S. citizen.     
  
  Note:  Additional eligibility requirements for specific designee types
  are included in the applicable designee type specific chapters
  (Chapters 6 and 7) of this order.


As to what the FAA is looking for (this from a DPE, I used to work with), the FAA is looking for someone they can trust to perform the job to standards. They expect a DPE candidate to be an experienced instructor.  They'll look at the the history of students signed off for check rides by that person. If any significant number of recommended students aren't passing check rides or 'just squeak by', it won't reflect favorably on the applicant.  OTOH, if the recommended students are well prepared and almost always pass, that's a good indication. That shows the instructor's standards are the same as the FAA's.  

'Openings' for DPE positions are driven by supply and demand. The purpose of any Designee is to help manage the workload imposed on the FAA's office. If the office can handle the number of applicant's, they won't add a DPE. And if an existing DPE isn't performing a minimum of examinations, they'll lose the Designation. Most of the DPEs I've known worked for flight schools where there was a steady stream of students.