Audio Processing Used in Voip Mobile Apps
How would a VoIP app developer decide what audio processing should be
  used in an app ?  
  


experience, testing, looking at what others do (aka. typical software development).

So, you, for example, know that you need to work in situations where you'll have echos of your speaker in your microphone, so you add echo cancellation. 

Do that with all negative effects that you know beforehand.

Then build a prototype, identify problems, find method to solve problem, repeat.

All in all, engineering.


  
  Is there a list of all processing options somewhere
  online ?  
  


Sorry, that list would be infinite, so no.

But: You could read a (good) book on speech processing. That would make (somewhat) of an expert on the topic, and experts find solutions to problems by applying their knowledge!


  
  Is there a methodology involved in selecting the needed
  audio processing for an app ?  
  


See first point. Engineering. 


  
  If I were to identify the minimum set
  of audio processing needed for a VoIP app, how would I go about doing
  it ?
  


I'd argue that you start with a decoder for the audio codec and call "none" a minimum amount of post-processing that. Then you work up until the application works as intended.