• Exactly as you suspect, it "re-rolls" the PID (and consequently ESV).
• - It has to do this at the time you pick up the egg. If it did it at the time you hatched, then the whole shiny hatching thing wouldn't work, since you'd check an egg's ESV, but it could still change when hatching.
• No, the parents must be similar.
- Assuming you have the Shiny Charm: then if you use non Masuda Method parents initially, you must switch to the same; if you use Masuda Method parents initially, again you must switch to the same.
- If you do not have the Shiny Charm then you must be using Masuda Method parents, and switch to the same. Non Masuda Method parents (without shiny charm) have a different PID each time you pick up the egg.
- The parents must also match on the number of possible genders. I.e. if you use a species with 2 genders (male/female) initially, then you must switch in the same. If you use a species with only 1 gender (always male, always female, genderless), then again you must switch to the same.