The following are the articles that are the foundation of my understanding of the topics:
http://bulbapedia.bulbagarden.net/wiki/PID
http://bulbapedia.bulbagarden.net/wiki/Breed
First off, I would like to address the following statements:
Trading doesn't change anything.
the game checks whether to generate extra PIDs
This is the primary focus of my confusion. These statements contradict each other. Which is it? Trading does not effect an egg's assigned PID 
OR does trading impact it?
Second off, I would like to address the following statements:
it might generate a different number of extra PIDs
the game checks whether to generate extra PIDs
You have intentionally these references in the plural. Is it not true that a single egg has a single generated PID value (set at time of its creation)?
The statements above indicates that instead of a single generated PID that instead the game generates X number of PID's then chooses one of those to apply? Is this a formula that used for all eggs or just traded eggs?