PID clarification

It's been a long while since I cared about PID's, thanks to Gen V, but I've gotten an interest in certain Pokemon from the past (particularly a couple of Events that are very unique)

So, I want to double-check how PID's work with different things, only applied to Gen III/IV.

-Nintendo Event (Does PID matter? If it does, which Encounter Method is used for Gen IV Events?)

--ex: Crown Suicune Event (Zoroark Movie)

-In-game Gift (Like with Events, does PID matter, and if so, what Encounter Method?)

---In particular, the Togepi from Pokemon , with it's unique Tri Attack.

-Hatching (from what I've read, Hatched can have any IV/Nature/Shiny/Ability combo, yes? If not, what's the limits?)