1. If I discard a dupe from Kings Landing Joffrey, can it be canceled with He Calls it Thinking?

2. Can Kings Landing Joffrey's ability be canceled by He Calls It Thinking?

Pretty much it comes down to how Immunity works. I'm not sure if immunity covers abilities on characters since it apparently doesn't cover power on a character.

Staton said:

(3.19) Scope of Immunity
Immunity only protects a card itself. Peripheral
entities attached to or associated with a card,
such as attachments, duplicates, power counters,
gold tokens on the immune card, and also
including triggered effects originating from the
immune card, can still be affected by cards of
the type to which that card is immune, as long
as the affecting cards do not target the immune
card.

~ Man, that FAQ thing is pretty useful sometimes, huh?